Train from Geneva Airport to Fiesch

I don't see any reason why you wouldn't be able to. There are both ticket machines and manned positions in the Geneva Airport train station. As long as you have cash or an acceptable card it should be quite straightforward.

There are two trains an hour, so even if you have to wait in line a bit it shouldn't be an issue.

However, I don't see a reason either for not being able to buy the tickets online. Have you tried sbb.ch?

Trains in Switzerland are mass transit. Used by normal people to go about their everyday business, as well as by tourists. Having to buy tickets in advance, and having to reserve seats would not be accepted by the people.

Therefore, just like with trains elsewhere in Europe you can always buy tickets at the point of departure. And in Switzerland most trains don't even offer the possibility to reserve seats.

SBB sells tickets on line and you can do this up to 60 days in advance. However most tickets are sold mere minutes before departure, and increasingly people are using their phones to do so, and for that reason SBB is even starting to reduce the number of ticket vending machines.

The tickets most people buy are normal flexible tickets, which allow you to take any train on the route you bought, and allow stopovers. About half the Swiss have a Half Fare Card, as that quickly pays off if you use public transport regularly.

Buying a ticket in advance does give you the opportunity to save some money. There are two interesting offers here:

• There is the "Day Saver Pass". This is a pass that allows you to travel freely on the whole public transport network for a whole day. You can only buy this on line, and it gets progressively more expensive closer to the travel day. This is a good option for someone arriving by plane, as it is fully flexible:https://www.sbb.ch/en/travelcards-and-tickets/tickets-for-switzerland/1-day-travelpass/saver-day-pass.html

• There are so called "Super Saver Tickets". These are also available from 60 days out, and offer reductions up to 70%. But these tickets are not flexible. This is however a good choice for long distance trips within Switzerland, and for trips to the airport, as the savings can be substantial. Just book these on line (or using the phone app). They are offered automatically when available. With this type of ticket you do need to take the train you booked, unless you cannot reach it due to a fault of the transport company.