Tegel Airport to East Berlin

There is no rail transport to Tegel airport. The most direct through route to East Berlin is the TXL express bus to Alexanderplatz. If you are staying in the northern part of old East Berlin, it may be quicker to take the X9 bus to Jungfernheide station on the S-Bahn circle line, or the 128 to Kurt Schumacher Platz U-Bahn station on line U6.

I've taken the TXL bus to Alexanderplatz and to Hauptbanhof, it's very easy and runs about every 10-15 minutes during peak times of day, sometimes more often than that.

There are buses as stated above, or you can take a taxi (which shouldn't be more than €30). Either works fine - your choice.

http://www.berlin-airport.de/en/travellers-txl/to-and-from/buses-and-trains/index.php

Which part of East Berlin are you staying? If you are in Prenzlauer Berg, you can take the bus to Beusselstrasse and then the S-Bahn to Schonhauser Allee or Prenzlauer Allee. You can go a bit further to Storkower Strasse if you are going to Friedrichshain.