Old watches had to be wound up when they ran out of power.The person is comparing himself to an old style watch.The person is saying they need Sunday to rest and renew their energy.They like to relax on Sundays and not get all "wound up" or full of tension.

I'm reading 'Norwegian Wood' also.

I'm reading 'Norwegian Wood' in 2020 🙃

I got here by Murakami too :D