From the airport you can take bus 22, that goes to the city centre and then take trolleybus 8 from Grecinieku iela in the Old Town that goes directly to the hotel; your stop name is Gimnastikas iela.
Trolleybus no. 8 - terminal Central Station
Bus no. 42 - passes Central Station; terminal Central Marketplace and Central Bus Station
Minibus no. 233A - terminal Train Station and Central Post Office.