Can smartphones power a MCU circuit through an NFC tag?
Every NFC tag is an MCU. (So, that answers your title: yes.)
The Attiny is probably not the tool of choice here. There's NFC devices meant for pretty much what you describe: changing what is displayed.
When you look at a few larger stores, you'll find they have e-paper displays that show the price of the wares on the shelves, and these things are wirelessly updated. I don't remember whether they have a secondary power source (e.g. a coin cell) to help change the display – either way, that would be good enough for the application you describe.
So, look at e-Paper in-store price tags that are wirelessly reprogrammed. Pretty much what you want.