Is it possible to buy a 24-hour ticket at Copenhagen airport in advance?

No. Unless I misremember, the 24-hour tickets used to be cardboard tickets with the same format as the 10-punch strip tickets, which you would validate when first boarding a bus or train. But these strip tickets were discontinued several years ago in favor of contactless stored-value cards, so the mechanical validators have been removed from buses.

Your options are:

1. Paper 24-hour tickets can be bought from vending machines at any train/metro station, so if you're staying somewhere reasonably central you may not need to buy it in advance.

2. On Android or Iphone, download the app DOT Mobilbilletter linked from http://dinoffentligetransport.dk/service/for-tourists/tickets/. The app lets you pay with a payment card and does speak English. You show the ticket on the phone screen to bus drivers and ticket inspectors.

3. Finally it can be bought as an SMS ticket, paid for by replying to an overtaxed SMS message. The transit agency promises this works with foreign phone numbers, but the instructions at http://dinoffentligetransport.dk/hjaelp-til-billetter/salgssteder/apps-og-sms/sms-billet/ and http://dinoffentligetransport.dk/billetter-og-priser/find-og-koeb-billet/koeb-billet/ only seem to be available in Danish, so you may need to get someone such as a hotel receptionist to help you with the procedure.