My division's main office is on the corner of Church and 4th and I go a few time a month, sometimes stay overnight. As far as current travel conditions, look at TDOT online and some of the Nashville papers online. Last week there was a section of I-24 that was closed due to the road damage from the flood. They are working hard to get things reparied and functioning again, but some places like Opry Mills Mall and the Opryland Hotel will take some time due to the extensive damage and I'm betting all of the road damage from them being undermined has not surfaced yet.
As far as budget hotels- depends on what you call budget. You may be able to get a decent rate at the Homewood Suites by Hilton on Church Street, that's where I generally stay if I overnight for work. They have a really nice breakfast with a hot item, breakfast buritto and eggs and bacon are examples, plus the standard continental fare including a waffle maker and if you are there during the week they also have a dinner included with the room rate. That should save you money on food. Holiday Inn Express generally my second choice. It is on Broadway and has an expanded hot continental breakfast. The Renissance is really close to TPAC, pretty pricey but some packages include a couple of free breakfast buffets, my hubby and son said it was great. I missed it because I had to go into work. Also most all hotels downtown also charge for parking.
Any hotel you call, and I recommend calling any hotel directly to check on rates as well as online because they may have a unpublished special, be sure to tell them you would like to stay in the city to help the local economy after the flood, but you have a limited budget. You never know what rate you might get.
Good luck and have fun!!