A Tiny85 is massive overkill but it will work if that's what you can get. I typically use a Tiny13a for simple stuff like this, they were the cheapest AVR I could find at the time I bought a handful of them.
It's possible to get the standby power down to microamps, I have some battery powered projects that get well over a year from a CR123A cell. You have to run directly from the battery though to do that, the quiescent draw of a typical linear regulator will dwarf what the micro draws when idle. The key is to have it spend most of its time sleeping and then use a timer interrupt to wake it up periodically.
The Arduino ecosystem supports the Tiny85 directly, and there is a way to use the smaller AVRs too if you prefer to use that instead of C.