Pity no one has tried to help out a beginner. While I know little of Unity, I can offer some C# comments.
Let me start with that overall your code looks nice. It has good indentation and variable names are meaningful, i.e. not cryptic or abbreviated. Use of braces is very good.
Actually, I don't see any properties in your class, but that might be a ...
What you're actually doing is implementing Service Locator which is quite opposite to dependency injection. Please refer to [THIS] discussion about it.
Please use already existing solution.
Few points from my side, just to point out why I think that's wrong for your case:
Why do you think you need more lightweight DI container? Have you performed ...