Riga is a very walkable city. The historical Old Town is fairly small and the Art Nuveau district within walking distance.
We stayed at Mercure close to the railway station. There are other options including Radisson.
Both FR and BT fly from ARN and have competitive fares.
I recall from the airport we used public bus that goes to the railway station. Tickets can be purchased from the newsstand - you can communicate in English without too many issues.