There is a Fairfield Inn on the same grounds as the airport Marriott. This will be closer to your budget expectations. Avoid the MIA Airport hotel, not really clean, badly needs renovation. The five minutes in the shuttle to the Marriott / Fairfield is worth it to avoid the airport hotel.

Also keep in mind, that if you do not want to wait for a hotel shuttle, you must use the special local cabs, not the general cabs. There are two different cab services at MIA, one for close in trips and one for trips further out. If you have any doubt, tell the dispatcher where you are going.