The National Archives of Estonia is located about 2,5 km from the city centre which means about 30 minutes walking only. If you prefer local transportation, here are some tips.
Taxis are relatively cheap, meaning that a ride to/from the city centre would cost less than 10 €. People at the hotels are certainly more than glad to order you one.
Multiple local buses go directly between the city centre and the archives. The main bus stop is called Kaubamaja there you can take the bus No. 1 (to stop Soinaste) or bus No. 3 (to stop Tehnoloogiainstituut). A Google Map to help you is here.
Bus tickets can be bought in the buses (1,5 €) or from any “R Kiosk”. In the last case you will have to buy a transport card for 2€ (or a sticker which you can attach to your wallet or phone) and load some money onto the card. A single ride costs 0.83 €. More information on local buses in Tartu can be found here.
A Google Map with the main locations is available here.