Monostable circuit with 1 hour high output
Use a CD4060 instead by itself - it has a built-in oscillator so you don’t need the 555. Then gate the circuit to get the one-shot behavior you’re looking for.
MORE: To get an accurate time, use two CD4060s.

use a 'watch crystal' with the first one to make a 2Hz signal
use the second one to count down from 8Hz (more on why, below.)

Here's the first part to make the reference:

From this Q: Calculating Rs Value for 32kHz Pierce-Gate Crystal Oscillator Circuit
Now comes the second part. Take the 8Hz output (Q11) and feed it to a second 4060. Then that device has outputs that toggle as follows:

Now, you can decode count = 3600s, that is, one hour, as (2048 + 1024 + 512 + 16) off the second device when we see Q13, Q12, Q11 and Q6 = high.
Now, I explain why I chose 8Hz to the second stage: because I wanted to decode counts 2048, 1024, 512 and 16, and you'll see that Q10 isn't pinned out, so I scaled the counter bits so I didn't need to look at Q10.
This will give a 100's of PPM accuracy due to the watch crystal. I'll leave the rest of the design up to you, but I suggest using flip-flop with async set/reset to control MR and your device:

push button: set flop, Q=1=device on; Qn=0=MR, counter runs
decode count = 3600: reset flop, Q=0=device stops; Qn=0=MR, count stops