I was watching Good Night Sweet Heart the other night, and the Time Traveler, back in the present, finds appropriate clothing and old currency in a flea market.You would first though need to collect some old money in the present to take back to the future because how would even come up with 20 dollars other than to steal it. Which if you get caught opens you up to a big problem what with no ID and maybe the track down your past self. Though it might be better to just bring stuff from the future back and pawn it at pawn shops then use that money to go make low level bets. Of course all of this is time consuming. It might take you a year or more to build up enough money to invest. But still you then need to find a place to live. Hotels will ask for things like a credit card. If you want to drive you need a drivers license. Sooner or later you will need to bring in a guy or form a crew in order to get things done and a crew that won't ask to many questions.
Meanwhile on Journeyman, Owen from Grays Anatomy tried to pay for a Taxi with future currency, and 7 years later, the Secret Service finally tracks him down in the present, because that's their job, to track counterfeiters, even if they are hiding in the future.
On 12 Monkeys the tv show, they bought a room in a hotel in the 40s, and they stayed there, "whenever" they were in town before and after the apocalypse.