I tried something with using millis() to run a clock. That had a longterm drift (a minute or so every day). That was with an (older) Arduino with a crystals - the newers ones have something else that is not as regular.
Get an RTC chip, maybe on on a breakoutboard w/battery, and use it. (I dont know if you can get something that will interface easily with the ATTiny). By the way, I could set the 24 hour clock using just one push button (and some patientence)