If you're coming on Sunday and would like to check out the Terminal Tower observation deck and Tower City, the Soldiers and Sailors Monument, and The Arcade before the game, there are large surface lots just west of Public Square (the heart of downtown) that currently are $10 for all day on weekends. I don't know if they bump their fees for a game, but I doubt it given the distance from Progressive Field. See attractions in the "Cleveland Tips" travel article.
http://www.tripadvisor.com/Travel-g50207-c198218/Cleveland:Ohio:Tips.For.Visiting.Cleveland.html
These Public Square lots are about a walk of 10 minutes to Progressive Field. You can pick up a free bus trolley in front of the Horseshoe Casino on Public Square and ride it East 9th St. and walk a couple blocks south to Progressive Field. It would be more fun to walk down East 4th St. and turn left in front of the Q arena.
These lots typically are busy even into the early morning given their proximity to the Horseshoe Casino and the Warehouse District entertainment/dining district. If you're staying around downtown after the game, that also may be a attraction for these lots.
Another group of surface lots charging perhaps $5-10, depending upon the distance from the stadium (go an extra block and save $5) are on Prospect Ave. around E. 17t St. The Hanna Parking Garage also is there, but it charges $10 for games most of the time, especially if there are also events at the very nearby PlayhouseSquare, which there typically are on a Sunday in September. These lots also provide easy access to the Innerbelt avoiding much of the traffic around Progressive Field. To reach the stadium, walk west to Bolivar and take that to East 9th St. It's a 10-minute walk (the eight blocks are really about 3 normal city blocks). The big disadvantage of Prospect parking is that the walks are longer to downtown entertainment/dining districts and the lots will empty rapidly after the game.
On the Bolivar walk, note the Grays Armory building, one of the most historic in Cleveland. E.g., this was the first home of the internationally famed Cleveland Orchestra.
http://en.wikipedia.org/wiki/Grays_Armory
Sometimes, the Grays offer refreshments inside the building on game days. If so at the time of your trip, perhaps you can arrange a tour of the building/museum.
I'm not certain that there are parking garages charging just $5 on East 9th St. just a block north of Progressive Field. I doubt it. I used to park all the time at the garage on Chester just off East 9th St. It's three blocks north of Progressive Field, and the last time I parked there for a game it was $10. This garage has easy access to the I-90 Innerbelt avoiding the traffic after the game. Further north of East 9th St., perhaps around Superior, there probably are some $5 garages.
If this is an afternoon game, garages should be OK. If a night game, make certain you know the garage closing time.
Obviously, parking rates can change by the time of your trip.
Be certain to check out the excellent statue of Bob Feller, in his famed pitching stance, at the East 9th St. entrance to the stadium!
The back of the Northeast Ohio AAA map has an excellent map of downtown Cleveland. You also can download maps at positivelycleveland.com.
Enjoy Cleveland!