Can I make a shipping zone for just one city or one pincode?

Yes. In a zone's Regions, each row goes Country → State → City → Pincode, and each level unlocks once the one before it is chosen.

1. Go to Plugins → Shipping Zone and click Edit on the zone.
2. Click + Add region, pick the Country, State and City.
3. Leave Pincode as Any for the whole city, or pick one pincode. Add more rows for more pincodes.
4. Click Save zone.

A city or pincode zone wins over a wider state or country zone for the same address, so you can keep a general rate for the rest.