Can I book Gatwick Express at the airport?

You can buy it there, no problem. Or it is £2.20 less to buy online for a single (one-way) ticket. See https://www.gatwickexpress.com/tickets/buy-tickets.

absolutely. That is what we did and it was not problem at all

You have two conceptual misconceptions:
1) You do not "book" trains, you buy tickets. You buy a ticket and it is valid on any train, you then just go to the platform and get on the next train that leaves heading for London (Victoria station).
Depending on where your hotel is, you then need to get there (tube, bus, taxi, ....)
2) Customs is not a problem, you walk through the "nothing to declare" door and it takes less than 30 seconds. What takes time is disembarking, walking through corridors and the main cause of delay, immigration (passport control). Luggage pick-up and customs comes after that.

Gatwick has two terminals "north" and "south". The station is at the south terminal. If you are lucky enough to arrive there you just follow the signs to the train. If you arrive at the north terminal you have to get a people mover to the south terminal.

A rough guess is 60-90 minutes from when you land until you get on a train.

If you have a contactless card you can breeze through to the GE with that rather than joining the throngs of confused tourists.

Advanced purchase via the app can be cheaper than contactless card or Apple/Android pay, depending on various factors. Also note that it is much cheaper on the non-Gatwick Express services with contactless.

The entrance to the railway station at Gatwick can be incredibly crowded at times with people queuing for the machines / ticket office etc so anything you can do to avoid that (which includes collecting pre-booked paper tickets is worth consideration. (Doesn't bother me as my trips through Gatwick are returns, so already equipped ...).

I flew into Gatwick and immediately bought an Oyster Card, which I used (for a supplemental fee) to get from Gatwick into London. There are TI desks at Gatwick arrivals area where you can buy the card.

No scheduled train in Britain requires pre booking (except the overnight sleeping car services). On many it isn't even possible. It's just that a lot of longer journeys will be cheaper if you do, and a few will be full and you'll have to stand if you don't have a reservation.

Gatwick Express is a "premium" brand of Southern Rail, who also operate other trains. Both types go to Victoria station. Basic Southern prices are slightly less.

Thameslink are the cheapest but their central London stations aren't as convenient for Westminster.

How does the contactless payment system know if you've taken the expensive Gatwick Express or a local, cheaper train?

The Express comes into a dedicated area at Victoria so it is picked up when you leave.

Interesting question from Tom, and one which regular commuters know about. As Marco says, they have dedicated platforms and the card readers are basically set up as a different station. But sometimes things go wrong and the Express trains are routed into ordinary platforms, at which point the regulars tap out and disappear quickly!! But it can go wrong the other way, if an ordinary train is diverted into an Express platform the card users have to ask to be let through to an ordinary platform to tap out.

The contrast maybe is with Heathrow, where the Express have dedicated platforms but they are open, with all checking done on the train. No contactless complication yet though.