Can I charge a different tax rate for one pincode or city?

Yes. In Settings → Store Settings → Manage Tax, under Add a tax rate:

1. Choose the Country, State and City, then the Postal code (or leave Postal as *All ()** for the whole city).
2. Enter the extra Rate and a Name, such as Local Levy.
3. Click Add rate.

At checkout, every rate that matches the customer's address is added together. A customer in that pincode pays your normal rate plus this one, each shown as its own line. The hint on the page talks about the most specific match winning, but rates actually stack. So only add the extra amount here, not the full combined rate.