In the spirit of Half day in Berlin - after non-classical tourist spots I'd like to ask the community about what can be seen in Tallinn on a Saturday afternoon?
One afternoon is sufficient to see most, if not all, of the Old Town, which is part of the UNESCO World Heritage.
As an alternative, in case of bad weather, or as a complement, in case you are rushing too fast through the Old Town, you can consider a visit to the Museum of Occupations. Interesting, instructive and easy to reach:
Tallinn, if enjoyed slowly allows you to immerse in its energy, which is quite 'high' compared to the more polished neighbours across the Baltic owing to the low spirit rates. Hence you shall find lots of Finns and some russians hanging out regularly in Tallinn thronging its many pubs and night clubs, partying all night, now that the sun almost never sets.
For the day time, you have a couple of museums (naval, art, military, churches) etc and the charm of the old town can be slowly taken in if you wish to. You can always spend a lot of time with architecture and history if you are in those things. Else have a nice bite on some variety food, walk around, feel the energy, join the youth in their celebrations and catch some hot action at the night parties.