APU
Never flown a 737 but I would wager my months salary, lets make that my years salary, that the APU is in the MEL.
Speaking for my own type, the MEL states that one is installed and it is acceptable to dispatch the aircraft with it in an unservicable state.....(except for ER Ops with one exception. Depending on the rectification interval i.e B,C or D it can dispatch with a number of provisos including up to 138mins ETOPS dispatch with a u/s APU).
Starting engines is then just a simple case of connecting external pnematic power and then once the engine is safely turning disconnecting the external power and doing a crossbleed start. Can take a while to cool the cabin down in these circumstances unless a/con has been connected during the turnaround.
At the other end it's very important to ensure that the GPU is plugged in before stopping the engines on stand ( which is why you sometimes pull onto stand and sit for a few minutes with the seat belt sign illuminated and twiddle your thumbs wondering why. I tend to let the pax know about this beforehand).