How should I go about replacing a cracked sanitary tee in a sink drain?

Yes, you'll simply cut all three pipes back a ways and rebuild using couplers, as you suggested. I would do the vertical joints first, cementing them all at once so you can twist them well during assembly. You should then be able to flex it leftward enough to get the drain parts in place. However you go about it, have a plan. Cemented parts don't come apart after about a minute.

Note that the tee probably failed due to stress. Unless it's a trick of the light, there appears to be a slight angle to it, possibly due to being torqued or pulled by the incoming drain line. Be sure all your parts fit comfortably before and during cementing.

You've got it! Make sure that you

• use matching black ABS pipe, not white PVC
• get glue appropriate for ABS, not PVC
• make sure you clean off the drain pipe so the glue will adhere properly
• since you've got a fair bit of room to work, cut your straight pipes far enough back that you can easily work with them
  • do not try cut right at the edge of the T, your life will be miserable.
  • I'd suggest cutting the drain just enough up from the sole plate that you can slip the coupler on with a bit of room to spare.
  • Cut the horizontal about mid-way between the studs
  • Cut the vent high enough that you've got room to work, but not too close to the extra stud - it might interfere with getting the coupling on.

You might need to use a flexible coupling to get the T reinstalled since you've got to get all 3 in at the same time. OTOH, there may be enough flexibility in the vent stack that you can wiggle it together without one - you may have to play with it as you're dry fitting everything to find out.