IMHO, if you are not an EE, then I think it will be harder without a small microcontroller than with.
Or you could go here:
Forrest M. Mims III
And use that information to do some lookup on his "Engineer's Mini-Notebooks", buy copies of all of them (they should be required reading for Arduino newbies), and find the circuit in there to do what you want with a minimum of headache.
Here's my $0.02 worth.
If your near term goal is to learn about electronics, you could give Forrest M. Mims III and publisher $19.95+10*$12.95 average (10 Engineer's Mini Notebooks that I saw at Amazon, but maybe more).
If you are quite 'academic', and like traditional text books, you might also consider The Art of Electronics. It is $89 at Amazon, and superb, but it not for everyone.
You do have a concrete project that you want to do, so it would be very a good focus for learning. You have understood the problem, and have a working solution.
If your near term goal is not to learn lots about electronics, but just enough to make your project, then you could go along other paths.
For example, you could exploit the fact that an ATtiny is significantly more controllable than most analogue electronics that you can build and debug unless you have test equipment, and ATtiny is probably lower cost. Also, you have many of the skills and much of the knowledge already. In fact you have a working project, which you might transfer to a lower cost device.
So you could spend a few $ on experiments with an ATtiny, use your Arduino to program it, and also develop useful skills. Then choose.
The nice thing about an ATtiny-based approach is you could change some of its behaviour after the electronics is built, by changing firmware. So, for example, you might want the light to come on for only an hour after dark. That would be quite a lot to add using non-programmable electronics.
Up to you.