If not, then there is no way that Armory will be able to show you your bitcoins (and therefore it won't let you spend them either). Armory reads the Bitcoin Core blockdata to see transactions that relate to your wallet... and it uses the Bitcoin Core RPC system to send out transactions.
If you don't want to have to wait until Bitcoin Core is fully synced (will likely take a day or three, and require something like 280gigs of harddrive space/bandwidth)... then you other option is to export the private key for the address that you sent the coins to.
You can find the private key by following this old guide of mine: https://bitcointalksearch.org/topic/m.43255691
I would recommend that you download and install Electrum from here: www.electrum.org/#download
Create a new wallet in Electrum and choose the "Import Bitcoin Addresses or Private Keys" option, then paste the private key that matches the one for your Armory address. You may need to prefix it with "p2pkh:" to make it generate the correct address in Electrum. So when you import into Electrum you have to enter it as: "p2pkh:YOUR_PRIVATE_KEY_GOES_HERE".
for instance: p2pkh:5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF
