Depends a lot of course on quality of construction and your locale, might get it for about $20/SF but more probably in the order of $40-50/SF including minimal ceiling lighting, but without any "floor" or deck - just basically a portico over the patio. You can find more prior questions with answers in the Home > Roofing and Home > Decks and Porches (or is it patios ?) categories in Browse Projects, at lower left.
Be sure to pay very close attention to how the new porch roof attaches to the house and provide fullice and water shield there - especially if you are in a snowy area, because improper valley construction or too long a valley might result in ice damming issues. The worst thing that people commonly do is make the porch roof a continuation of the existing roof so there is a very long roof section over the cold porch (or patio) to form ice dams, or worse yet making the porch roof flatter than the house roof it attaches to, so snow piles up at the interface.
Also, pay attention to guttering so you do not dump a lot of water right next to the foundation or where it can flood the patio - this is a case where double-ended gutters (downspout on each corner) all around can really pay off - with proper slope control it can be made so the ones closest to the house are slighly higher so they only flow if the outer ones plug up - and of course proper water control to keep it moving away from the patio and house.
Also, in an outdoor environment like that you might reconsider if beadboard is the right thing - due to moisture issues. Also, if putting in a ceiling be aware you need to absolutely close up the roof extension to prevent it becoming an insect, squirrel, bat, etc nesting area.
