Well I've only been the one time last year and as we had a car didn't look at the bus timetables at all I'm afraid.
Behind Cortina you have a load of routes on Pompagnon which you could probably walk to in an hour or so depending on fitness & sack weight...
Pompagagnon by
slack---line, on Flickr
If there is a bus running up/over the Falzerago Pass (and
this suggests there is) then you've got Cinque Torre, Col de Bois, Falzarego Towers and quite a few other crags at your disposal (Cofe has done one on Tofana I think, we did the Alveraza route at Col de Bois and a few routes at Cinque Torre which has sports as well as multi-pitch up the towers, several easy-ish lines on the three towers that are grouped next to each other which would make a good day out). You can get cable cars up to Cinque Torre (although if on a budget this may prove expensive) and there is also a cable car that goes from the road up to the higher parts of Pompagnon from the road that goes to Mizurina, although you'd still have to get there.
I'd say if you're looking at doing some of the longer multi-pitch routes you'd be hard pushed to do more than one in a day anyway unless really fast & efficient, so it would likely be a case of trekking out of Cortina each day (no idea what the authorities are like about wild-camping, saw a few people and stopped ourselves for one night on a back road, deer when we pulled up and something snaffled food leftovers from tent awning in the middle of the night).
One thing that might be worth considering, although its not really budget, is trekking up to huts and staying at those (having a specific BMC hut concession card would drastically reduce the cost, but if you do this and are offered any Genziana liqueur my advice would be to politely decline and have something with berries in instead).
Its a bit confusing, but in conjunction with UKC's map/overview in their crags database
Planetmountain.com has a lot of information on the Dolomites.
I don't think much of that is very useful I'm afraid, but hopefully others with more experience/knowledge can chip.