Active pickups should have a low output impedance and that will allow them to drive fairly high cable capacitances without any effect on tone. That is one of the things that a preamp will do for you too and if that is all you are after then there is no point to putting in an active preamp as well. Of course the on-board EQ controls that a preamp also gives you are a considerable convenience and if you think you would use them they are reason enough to put in a preamp.
The effect on battery life could be trivial or significant, it all depends on the current drain of the preamp compared to that of the pickup(s). A lot of preamps draw very little current and if the pickups draw a lot then there will be no significant loss of battery life. If the pickups and preamp both draw the same current then the battery life will be cut in half (although in many cases it will still be over a year), etc. We cannot give you a definitive answer about your battery life with the preamp because the devil is in the details which we do not know.