Puc microcontroller way over the top for me but feel free to post it i might consider making one myselfYour image is a bit difficult to see. Can you post a larger one?
Is this just for fun? Something you want to build on your own? Because there are cheap timers out there on eBay.
That battery is not going to last too long charging it that way. There is a whole science in charging batteries. You really should use a charge controller.
It looks like the light will be lite till the battery is depleted which is also very bad for the battery.
I have a similar project using a pic microcontroller. It monitors battery voltage and will shut the light off at a voltage that you can set. It is a bit over the top for the pupose but I can post it if you wish. But you would be willing to work with the pic micro.