The old city, as the name says, is the oldest part of the city. However it's very very small. Oldest in this case means until 18th century so apartments in this area might be quite small, with odd shapes, possibly lacking elevator. However, most of the city developed in the 19th and 20th. The type of buildings and accomodation varies a lot from quite old to very modern. A building might be super old and the next one might be recently renovated. So you'll have to check the hotel or Airbnb description in detail.

For easiest access on race day you want to be walking distance from theatre de verdure / jardin albert 1er.

Parking in Nice is difficult and expensive. Traffic can be very slow at times and promenade des anglais gets completely closed on race day. Unless you choose to stay in another town, you should and can do without a car.

If you want to consider options that are farther away from the main race area, you can easily move with public transportation. The tram lines are very frequent and efficient. Also, there's a bus service that follows the promenade.

Another option for moving around is the velobleu bike share system which has lots of stations. It's a bit finicky to use, the bikes are poorly maintained and dirty. However, it works quite well if you get the hang of it and allows very fast travel. There's also a few electric scooter sharing systems.

Keep in mind parts of Nice are hilly. There's some really nice Airbnb apartments on the hills. They are residential areas, very quiet with beautiful views. They are a brilliant option if staying longer but it might take a little longer to travel to the race area.