Unless you are a higher tier frequent flyer, it's going to be hard to get an easy (free) upgrade. Some airlines do offer paid last minute upgrades.
Eg, on Lufthansa, first they will go to LH Senators, then to Star Golds (non LH) and then to LH FTLs, then to other Star Silvers, then to plain old FFs and then to non FFs. By the time you got to LH FTLs, you probably would have easily eaten up the whole of business class!
For more info, goto www.flyertalk.com (bulletin boards) - best travel board, IMHO. Or try rec.travel.air