Prepare, fight the good fight, then cut and run.
Pre-open the supply crates, throwing them with the gravity gun so they break open and spill their contents without you necessarily picking them up. (nothing more frustrating than picking up a 45-shot ammo box and only getting 3 bullets out of it).
Set all the hopper mines you can find to guard the usual enemy hang-out points.
Then, work briskly between the two entry areas. Early on, before you're too badly mobbed, try to kill them on your side of the force field, so you can pick up their ammo without having to switch to the gravity gun. That won't be practical as the pace picks up.
Listen for Alyx's progress and as soon as it's possible, go blow the energy ball out of the generator and instantly run like the wind for the doorway Alyx just opened behind it. Seek cover, e.g. in that small control room on the left. Alyx will promptly close the door.
Taking a beating is hard to avoid. If you're hurt bad, choose carefully where to jump down to minimize fall damage. The next few areas won't contain much in the way of enemies, but have a goodly amount of supplies if you search. Look behind plain boxes (including on high shelves) and also catch the Lambda cache at the bottom of the damaged stairwell at the bottomless pit.
If that's not enough, fight carefully through the zombie-headcrab area that's next, making full use of the barnacles to kill enemies for you. The area with a poison zombie has two Lambda caches, so take the time to kill the black crabs and zombie, then explore.