ASTRA FLT CREW CLIMBED PREMATURELY FROM 1500 FT TO 2000 FT ON THE TEB SID.

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WE WERE DEPARTING TEB. WX CONDITIONS WERE WINDY AND COLD WITH LIGHT RAIN. CEILINGS WERE APPROX 3000 FT. ATIS WAS CALLING DEPS OFF RWY 19 AND RWY 24. THE WIND WAS FAVORING RWY 19 SO WE BRIEFED THE DEP FOR THAT RWY. UPON TAXI WE WERE ISSUED RWY 24 AS THE DEPARTING RWY. WE BRIEFED THE DEP PROCS AS SOON AS WE FINISHED THE TAXI CHKLIST. THIS WAS JUST PRIOR TO ARRIVING IN THE HOLD BOX AT TEB. WE SAT IN THE HOLD BOX APPROX 10 MIN WHILE WAITING FOR OUR RELEASE. UPON DEP THE CO-PILOT'S AIRSPEED INDICATOR SEEMED TO FAIL AS IT WAS FLAGGED AND THE NEEDLE MOVED ERRATICALLY. THE SID CALLED FOR A RWY HDG CLB TO 1500 FT FOLLOWED BY A TURN AND THEN A CLB TO 2000 FT AFTER 4.5 DME. I CLIMBED TO 1500 FT MADE THE INITIAL TURN AND THEN CLIMBED TO 2000 FT. I REACHED 2000 FT ABOUT .5 TO 1 MILE EARLY. 2000 FT IS WHAT IS CALLED FOR AFTER 4.5 DME FROM TEB. AT THIS POINT ATC ADVISED US TO PLEASE REVIEW THE SID CLOSER IN THE FUTURE AND GAVE US A CLB TO A HIGHER ALT. IT WAS AT THAT TIME WE REALIZED THE DEVIATION FROM THE SID. WE APOLOGIZED FOR THE DEVIATION AND SOON AFTER WERE GIVEN A HANDOFF TO THE NEXT CTLR. CONTRIBUTING FACTORS WERE: 1) OUR UNFAMILIARITY WITH THE SID. 2) THE DELAY AT THE END WHILE WAITING FOR OUR RELEASE ALLOWING TIME TO FORGET SOME OF THE DETAILS OF THE SID. 3) THE FAILED AIRSPEED INDICATOR AND OUR ATTENTION ON IT. IN THE FUTURE I WILL TAKE MORE TIME TO FULLY BRIEF THE SID ESPECIALLY ONE THAT I AM UNFAMILIAR WITH AND IN THE EVENT OF LONG DELAYS PRIOR TO DEP; TAKE THE TIME TO REVIEW IT PRIOR TO DEPARTING.