OK - now I understand better, so I will try to take you one step further. Abbreviation following - LF = lineal foot, LB = pound, CF = cubic feet, CY = cubic yard = 27 cubic feet, $ = pain in the pocketbook.
I presume this parallels the public roadway, but even if not, no difference other than restrictions on culvert size and placement will be less, and not require an engineers signoff. This is clearly an established drainage ditch because of the culverts on both ends - sounds like they are black plastic pipe, probably ribbed on the outside (and maybe smooth or maybe ribbed inside) like this - http://www.roperld.com/science/graphi...
You cannot block the neighbors' drainage, either ethically or legally, so you would need to get a permit from the town (or maybe county - depending on local setup) to install a connecting culvert with the neighbor ones, then fill the ditch over the top. Be sure to check who owns and maintains the road to get their permission, and if you have a local drainage and water commission or board they may have to sign off even if not along the road - check with your local county and city planning department (and state, if a state road).
The grant you heard about may be a road safety grant to promote filling deep roadside ditches and replacing them with culverts, but I suspect what you heard may have been misunderstood by the person who told you. Commonly the local road department will share cost or even do the install themselves if you provide the culvert, to replace swale type ditch crossings with culverts but that is only at driveways, so you would have to contact your road department about that. Another possibility is your local mosquito abatement board may give grants to replace ditches with culverts, but I doubt that.
The culvert has to be bedded in compacted bedding material (typically 1" minus crushed rock) to typically about 6 inches all around it to prevent crushing, then general fill to grade with compaction in about 1' compacted lifts (assuming you never intend to build any structure on it). The general fill should be graded smooth to conform to your desired final shape, then the topping finer fill should be graded smooth and planar and compacted also, to form a firm non-settling bed for your topsoil, which you do not compact or do only by hand with a rented lawn roller.
Now for the cost - the plastic culvert (I assumed 12-18" required by county/city) costs about $10-15/LF, so that is $500-1500 depending on diameter and length. Installation in bedding material about $5/LF - so another $250-500 there. Then the fill to grade - figure your cross sectional area, multiply by about 1.15 for compaction, times length, divide by 27 and that is your delivered volume, roughly. Say assume, just as an example (you can substitute your actual values later), a middepth average width of 6 feet and depth of 5 feet would be about 30SF per LF, times 1.15 times 50 LF / 27 cubic feet per cubic yard = 64 CY for a 50' length or 128 for 100 LF, with a top width of say 12'. Therefore, assuming the general fill will be coarse (say 18" minus) and varied, assume a 6 inch topping or levelling course of 2" minus gravel or crushed rock = 0.5' x 12' x 50' / 27 = 11 CY for 50 LF of ditch, 22 for 100. Topsoil say 3" thick = half that volume, or about 6-11 CY depending on length.
If along the road, assume permit and civil engineer's culvert spec costs of about $200-300, though county/city may do the engineering for free or by a standard specification sheet.
Using the price ranges I gave you before for the general materials, that works out to (at the extreme cost ranges - using lowest prices for 50' case and highest for 100 LF to give the extreme cost range), from $750-2000 for the culvert installed, $200-300 permit and engineer, 64-128 CY general fill at $5-15/CY, 11-22 CY at $12-20/CY, and 6-11 CY topsoil at $20-30/CY, plus say about 2# grass seed/100 SF at say $2/lb and about 40# fertilizer/1000SF (over first year) at $2/lb gives a total of about $1594 at the low end for a 50' run and lowest materials costs, to $5134 at high end for 100LF and highest materials costs.
Remember, this was only using sample dimensions and estimated materials prices, but if you run through it with your own actual dimensions you can come up with your own range of cost, which should put you in good shape to actually get bids from contractors to see what your actual local costs will be. This is a fairly small job, so you will likely get your bet price from a 1-2 man working owner general excavation outfit, though if there is a contractor doing road work or another driveway and culvert in the area, he might give you a good rate to combine the two jobs.
I would suggest you google a few articles and look at a couple of Youtube or manufacturer's videos on how to install culverts, how to compact fill, and how to plant a lawn (which I have assumed you will do yourself to save cost) so you have some familiarity with the process. The most important things in your job (other than having proper approvals so some government agency does not make you dig it back out and redo it), is proper installation of the culvert, straight even grade, the supporting bedding fill, getting the splices to the other culverts right, and then not running heavy equipment right on top of the pipe before it has a foot or two of cover over it. A light dozer or sidewalk size roller or hand-pushed compactor is fine once the 6" bedding is done and a foot of fill is over it, but heavy trucks or dozers should not be over it until 2-3 feet of cover has been placed. After the bedding is around it to the top but before the bedding is placed over it, it should be eyeballed to be sure it is straight along the top (at a uniform slope from one end to another). After the first couple of feet of fill is placed, you should get down on your hands and knees and look through the culverts (hopefully the adjacent ones are straight and round so you can see through) to make sure it is not bowed significantly or crushed - this is the time to dig it back up if they crushed it in early compaction.
Be sure to get your permits first to be able to provide them and any accompanying specifications to the bidders, because the road department undoubtedly has specific size minimums for the culverts, splice requirements, bedding dimensions and maybe material specification, limitations on total length of culvert without a cleanout, etc that you need to consider BEFORE working up your scope of work for bidders. The engineer can help you wiht these - most road and drainage departments and boards have handout sheets with this info. Also, while it will be cheaper to get all the fill done at once, the fill will settle some over the first year unless you pay an extra $2-3/CY to have them compact it as structural fill, so if you want a really smooth lawn you want to wait on doing the lawn till the second year - you could have them place the levelling course and then have them leave a pile of say 2-3 CY of material for you to use for levelling next year, and they could dump the topsoil in a couple of piles for you to then place yourself next year after levelling - a good few days a week evening exercise for you and the kids.
One other thing - if this cost is too much for your budget, then contact your state or local mosquito abatement board or drainage commission, and check with the state Copperative Extension Service, for ways to reduce insect habitat in ditches. Basically, raking out rocks and putting in a bit of fill to smooth it out and give the bottom a uniformm slope between the adjacent culvert so you can mow it would be a cheap solution. If you keep it mowed short it will not be significant insect habitat. Obviously this is a lot more work for you, but probably a truckload of dirt would allow you to level out and grass a normal 50-100' ditch for about $500 or less.
Good luck with your project.