Taxis are no problem at all.
The airport is only about about 7 miles from downtown. Traffic from the airport can be crappy if you travel during rush hour.
As far as where to stay, I'd first recommend the Union Station Hotel. It's a historic property that used to be a train depot many decades ago. It's located downtown, and not too far from the Ryman. An alternative that's also downtown is the Hermitage Hotel, another historic property. My final suggestion may be obvious, but it's the Opryland Hotel. Although it isn't downtown, it is something to see. Plus, if you want to go to the Opry, it's "next door".
If you make the trip, hope you and your brother have a good time.
