I dont think you can achieve it using Mist, because the application is structured in such a way where it has to wait for the Blockchain to sync to latest stage before it even starts.
There are online wallets which helps you transact using private keys; like myetherwallet.com
However, light clients are coming in the next Ethereum release, Metropolis. A light client is exactly what you're describing. A wallet or Ethereum (Dapp) browser that enables users to browse, receive and send transactions without having to synchronize the entire chain on a full node.
Meanwhile, you can use any Blockchain explorer like etherscan.io to see your balance by entering your public key.