They said it takes about 10 minutes to verify a bitcoin transaction. Does this mean if I purchase a bitcoin I have to wait 10 minutes for this transaction?
Yes if your transaction was lucky enough (or expensive enough) to make it into the next block. Since Bitcoin's 10-minute blocks hold a finite number of transactions, you might need to wait longer than that if your transaction wasn't included in the upcoming block.
On top of that, once your transaction is on the blockchain, you should ideally wait until several more blocks are mined until you're confident that your transaction got included in the globally-accepted branch (the longest branch).
So, to be safe, if you decide to wait for 6 blocks after your transaction is included, you'd end up waiting an additional hour until you're relatively confident that your transaction has been accepted.
If yes, how can I handle the real-time system requirement in the blockchain?
That's one challenge that Bitcoin hasn't overcome yet. The closest work-in-progress solution is the Lightning Network, but that's still in development.
In the meantime, some applications might decide to process real-time transactions off-chain by accepting transactions before they're included in the blockchain. But this is a risky trade-off; you'd allow fast transactions at the cost of security, opening yourself up to double-spending.