What is the recommendation to catch an early morning flight from Mitte Berlin to Tegel before the buses start running?
Are you sure you can't make it with the bus? Legally (noise regulations) planes can't start before 06:00 in Tegel. The first bus of the TXL line leaves Alexanderplatz at 04:21 and arrives at Tegel airport at 04:49. If that's too close to comfort, you'll have to call a cab.
A taxi ride at that time of the day would be €20, €25 max from Mitte to Tegel - if that's of any help.