The first guidance you need is from your contract or apartment manager, as to what kind of hanging systems you are allowed to use, and which ones are going to cause damage that will be applied to your damage deposit. Normally, any kind of adhesive hanger that is not totally cleaned off, and any but the smallest picture hanger holes (and sometimes even them) is going to ding your damage deposit unless you repair it before moving out.
The normal charger for a handyman will be in the $40-80/hour range (commonly with a $75-100 minimum), depending on his skills and labor costs in your area. Simple hanging of art, hanging mirrors, etc takes from 1-5 minutes per item typically - there are very simple commercial hangers for this. It is likely that he will have to scope out the work and then go get specific hanging systems for your needs, so count on about an hour of up-front time for that in addition to actual hanging time.
A mirror that is not hung by a bracket or cable on the back is tougher - might take 1/2-2 hours depending on the construction and how it is designed to be hung. Many flat panel mirrors (like bathroom mirrors) are designed to be hung by using very aggressive adhesive to the wall, and cannot be removed without breaking the mirror. This type of adhesive would definitely damage the wall and ding your deposit, so the handyman might have to come up with a wood or metal outer frame (like a large picture frame) to hold the mirror, and then hang that from heavy duty picture hangers.
By light fixtures, if you mean free-hanging on the wall or ceiling (again with the manager approval) with wall socket plugs, same as hanging a picture. If you mean changing out light fixtures for what is there now, a handyman can do that too. If you need new wiring and box then I would recommend a licensed electrician do it, and of course manager approval, and you will probably have to replace your fixture with a new suitable one if you take yours with you whenn you move.
