Let's just get this straight. The first rule about on-airplane skincare routines is you shouldn't be doing them. What you should do instead is an intense skincare routine before you board, followed by another when you land.
Why shouldn't you do your skincare on the plane, you ask? Exposing your bare skin to recycled airplane air is never good. Essentially, the air on a plane is dryer than normal air, and will dry your skin out more than normal air would. If you board the plane and then do your skincare routine, you've already exposed your bare skin to that air and it's going to result in your skin feeling like the Sahara. What you really want is to have already applied your skincare so it acts as a barrier between your skin and that dry, dry air.
What you should also do is up your water intake a couple hours prior to boarding. Yes, you'll have to use the bathroom on board and we all know how traumatic that can be, but it's for a good cause.
Here is my suggestion for a pre-boarding skincare routine!
When I fly, I essentially treat my skin like it's nighttime even if I'm flying out at 7am. So double cleansing is a must. To do that, you just need to cleanse with an oil-based cleanser then a water-based one. I'd then follow that up with a toner or facial spray. After that would be a perfect time to use a hyaluronic acid serum, as they work best when they have a bit of moisture to work with. The you should seal that moisture in with a good, super occlusive, moisturiser. So one you would normally consider a 'night time' one.
When doing super long flights (hello, Australia to anywhere), I actually like to layer a few moisturisers. When I flew from Australia to Canada, I layered Go-To Very Useful Face Cream with a bit of Weleda Skin Food. It was a great combination. If you're going to be flying at all during sunlight hours, now is when you should put sunscreen on. It's important to wear sunscreen on planes. While the windows on planes block out UVB rays so you don't burn, they don't block out UVA. Those are the ones that cause ageing. Yeah, wear your sunscreen. The last magic ingredient I'd suggest would be a facial oil to truly make sure none of the moisture in your skin is going anywhere during that flight.
Once you arrive at your destination, I'd suggest doing this routine again with an added chemical exfoliant thrown in to thoroughly clean everything out. And I wouldn't say no to a face mask, either.