Omniva (formerly Eesti Post) is the national postal service provider in Estonia. On their webpage you will find all information related to opening hours (which can vary greatly from one place to another) and postal codes.
Also, you can find mailboxes all around. They are orange and can be found in shopping centres, at most gas stations and outside post offices. You will see the pickup times of each box noted on them. If you need stamps, you will be able to find them at kiosks, post offices and grocery shops.
Post offices in Tartu city centre
Kvartal shopping centre
On this map you can check the location of other post offices and parcel machines in Tartu, as well as opening hours.