Downtown prices are high -- you're smart to look elsewhere! Airport or even a little farther out in Hermitage are near enough. Don't use the cheapest one, you get what you pay for and that includes security. Anywhere in Nashville will have Uber/Lyft to get you downtown. You could also stay nearer to Vanderbilt or south around Brentwood. None of these would be party central, ha! That said, none would be historic, either. If you want a truly historic hotel go with the Union Station Nashville Yards or the Hermitage Hotel. I think the Bobby is also historic, but is new from old, if that makes sense. It's near Printer's Alley. These would be splurges.
A must see is The Parthenon, a full sized replica of the Greek Parthenon--but no crumblies! It's in Centennial Park near Vanderbilt University--it's lit up at night, but there is a huge statue of Athena inside if you want to go in during the day. Just walking down Lower Broad and looking at the architecture vs the neon will give you a good idea of the original style. Listen at the door and if you like the music offered head inside--we don't JUST have country! Lots have rooftop bars-- the one on Acme looks over the Cumberland River. There are all sorts of architectural styles in the city. Most prominent is Greek but there is also a Gothic cathedral (Wightman Chapel) and the only existing church featuring an Egyptian style (Downtown Presbyterian)
In the downtown area, the Country Music Hall of Fame is a good place to visit, as is the Ryman, though I found the overall tour of the Ryman to be a little pricey -- you can get a pretty good feel for it by going to the gift shop and taking a peek into the auditorium without the actual tour. I like Hatch Show Print--it's a working museum. Frist Museum is an art deco post office turned into a museum--you can just look at the building without paying to see the exhibits. It is right next to Union Station, our old train station. Historic RCA Studio B is not in the general arena area but is downtown and worth a visit.
All in all, you're not going to have time to do everything so just pick what interests you the most or the area you like most and enjoy the trip. Go Preds!