The very first date that can be ascertained from the movie is December 1st, 1982. This date is written on the paychecks that Winthorpe has the Duke brothers sign.
I'm not a Payroll guy, but I'm fairly certain that paychecks are dated on (or close to) the day they're issued. This means the checks were likely signed on November 30th.
Immediately following the paychecks being signed, the Duke brothers first bring up their bet. On Winthorpe's way out of the club, where the checks were signed, he encounters Valentine. Valentine takes Winthorpe's briefcase, which leads to his arrest. The Duke brothers then officially make the bet that leads to Winthorpe becoming homeless.
They bail Valentine out of jail the following day, which is assumed to be December 1st, and give him Winthorpe's house. Winthorpe gets arrested that same day and spends a night in jail. The following day, December 2nd, he meets Ophelia and begins rooming with her (thus, he is not truly homeless).
Backtracking for a sec, while in jail Valentine is harassed by two larger men, one of which says, "It ain't cool bein' no jive turkey so close to Thanksgiving", which lends weight to the December 1st date.
The next date officially brought up is Christmas Eve, when Winthorpe is sick in Ophelia's bed. In between, there isn't a single calendar or newspaper with a date that is legible, so no other dates can even be assumed.
Lastly, the whole deal with the USDA report takes place on New Years Day, and the report is set to be officially released on Jan 2nd (as stated earlier in the film). Therefore, Winthorpe is officially "homeless" almost exactly 1 month.
If anyone with a Payroll background can back up my assumption, I'd appreciate it.