As I haven't yet been online and the funds have thus not appeared in my armory wallet, will exporting the private keys still work?
Yes. You do not need to be online to receive Bitcoin. So long as your transactions have confirmed, then it will work.
I also tried sweeping the wallet with the bitcoin phone app by scanning the QR code of the 'paper backup', but it said it cannot recognise input.
Armory's QR code and backup stuff is completely unrelated to any other wallet and is Armory specific.
If this is a straightforward solution, I'd rather do this than have to download the blockchain, and would happily use a different wallet to armory.
It should be fairly straightforward. Follow the instructions I gave to get the private keys. Then you should download and install Electrum and sweep those private keys into your new Electrum wallet.