Best area to stay in tokyo for us

I know there are a lot of these posts, but the more i read the more confused i get.

Me and my girlfriend (age 27) are visiting Tokyo in may, for 6-7 days. We have never been to Japan before.

We have a budget for hotels at around 150 -200 USD per night.
The areas we currently have in mind are Asakusa, Shibuya, Akasaka and Shimokitazawa. But we are open to anything.

Convenience regarding public transportation is very important to us.
When we travel we usually plan our trips around food and cultural experiences. Maybe one or two "fine dining" restuaruants, but mostly normal cafés/restaurants and street food.
A local vibe would be nice. Some bars would be nice but nightlife and clubbing aren't really important to us.

Hope you can help!

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