The standard and the most cheap way (0,40 Ls or about 0,30 EUR) is to get there by bus.
You can find timetables on
Rigas Satiksme (Riga Transportation) website, choose "
Maršruti, saraksti" (lists of timetables), then "
Autobusu kustības maršruti" (bus timetables), the bus number is
22. Then choose the second transportation way (back -
Lidosta - Abrenes iela), click on the top bus station -
Lidosta Rīga. You will see the timetables for the bus going from the Airport to the city center,
darba dienas mean business days,
sestdiena, svētdiena are Saturday and Sunday. The first column in bold are hours, the second are minutes. As you may see, on working days the last bus goes from there at about midnight, on holidays at 23:40.
However if you arrive at the late hour, let's say 23:00 like in your case, the only way is to get to the city is by taxi, if you won't be able to catch the bus. There are a lot of taxi cars near airport, however it will be not that cheap. I suppose the price at least is 20 EUR.