Create new wallet on MyEtherWallet.
Save Keystore File. Save password. Back up externally (USB, paper).
Send from Exchange to the address you created on MyEtherWallet
Go to The DAO tab, unlock your wallet, and choose the amount you wish to send.
Done!
Help Page is Helpful • DAO Page
Alternatively, try to sync Mist. Read the top answer here on how to sync using geth --fast to hopefully speed along the process.
disclosure: I'm co-founder of MyEtherWallet. I only answered this because you mentioned that is what you were trying to do. I would normally recommend getting Mist set up first, but you are really cutting it close now. I still recommend that you get Mist set up, even if it's after you get your DAO Tokens. It is worthwhile, especially with all the Tokens and Contracts coming around.