Alas, not much time to do much. Maybe you can squeeze in a trip to Doi Suthep temple in the morning, and there's always the Night Bazaar. My favorite low-cost hotel of the moment is the
Thapae Boutique House, which should run in the $30 range. It's a couple minutes from the Thapae Gate and is about a 10-minute walk to either the Night Bazaar or the Warorot Market and its nearby fruit and flower markets.
Since you will be there in the evening, you might want to hire a songthaew and dine at the Palaad Tawanron restaurant. (Have the driver wait for you.) It's about a 20-minute ride from the Thapae area and is adjacent to the Chiang Mai Zoo, offering a very nice hillside view of the city lights. If you don't want to go that far, there are numerous riverside restaurants that are closer by and offer a pleasant dining atmosphere if the music isn't too loud. Chiang Mai is noted for its curries and sausages, so give them a try.
Transport from the airport is very easy. Go to the taxi desk. It's a flat fee of 120 baht to pretty much any hotel in Chiang Mai. Returning to the airport, any tuk tuk should do the trick.