I think I might vacuum the mattress well, add baking soda all over it and brush that in with a broom or brush. That coffee idea doesn’t sound half bad either. I like Febreeze, but it is $5. Another choice is Nature’s Miracle or a similar product that removes pet stains.
Then I’d let it sit for a week sealed up. If you’re trying to do this with stuff you already own, I’d suggest laying down a vinyl tablecloth or heavy shower curtain, then a quilt or comforter, then the fitted sheet. After a week, vacuum well and see how it goes.
I realize that this violates the “cheap” requirement, but if I had to stay with the used mattress, I’d seal it up with a zippered plastic moving bag, and buy a foam topper. They are about $150 at the discount store and quite comfortable. The used mattress would be reduced to the foundation for the topper and you could probably seal it up completely, eliminating body odor or dust problems.