I have been where you are right now. We took our inflatable mattress on a camping trip, and it would not blow up. We found that we had no means of powering up at the campsite, so we were left without any means of getting it pumped up. Eventually, we resorted to using a manual foot pump for a cycling bike, and while that worked, it took ages. It also did not provide the same level of balance as you would get with the in-built pump.
For me, the best way on how to inflate air mattress with built-in pump without electricity is to have a manual power source. For example, you could charge up a power brick or something that can store energy. You might even find that you can use something like your car battery or a set of extra-power batteries. Most in-built inflator pumps can be powered using more than just the main power source.
Do some reading into the brand that you happen to use, and you can generally find out a bit more about what kind of solution you would need. Learning a bit more about the product will give you an idea of what kind of power it needs to get started.
I also recommend that you look into this guideline on how to get your mattress inflating without power. They give you some great tips on alternative power sources and give you ideas on how to inflate the mattress manually. It might not be perfect, but it is better than being stranded without any kind of comfortable bedding to lie upon!
Keep this in mind, as it can save you from ditching your mattress ahead of time.