Aircraft with INOP APUs can generally be "air started" with a ground-based high-pressure air system. It's relatively common and I've been on a plane that had to do the procedure. It was entirely undramatic other than engines being started before the pushback, but I doubt most passengers even noticed.

Now, interestingly, the 787 is a "bleedless" aircraft, so it doesn't use high-pressure air from the APU to spool up the engines. I believe it can use its hefty bank of lithium-ion batteries to start its engines if the APU (and associated electrical generator) is INOP.

Not a pilot/engineer - just an enthusiast. Someone more au fait with the 787 might be able to correct me on the above.