Every thing I have read says you need to sync before upgrading but this will take 6 months plus.
I don't know where you read that, but it is false. It makes no difference when upgrading whether you are fully synced, not synced at all (as would be the case with a fresh installation), or anything in between. The new version should pick up where the old version left off and, since many optimizations have been made in recent versions, it will finish synchronizing much faster.
My questions are, can I backup my wallet.dat, update and not loose my coins without syncing?
Yes. You should in any case back up your wallet.dat whether you plan to upgrade or not.
Or, can I save my wallet.dat, download another wallet on a different PC and put my wallet.dat into that wallet?
If by "another wallet", you mean "something other than Bitcoin Core", then no, at least not easily. Bitcoin-Qt/Core wallet files are generally not compatible with other programs.