What is the cheapest graphics card for a desktop that can handle the new age of empires 2 definitive version and the Halo reach for PC? I'm thinking about getting new parts for my cooler master centurion 541 case and I would like to know the cheapest graphics card that can handle these games
Halo Reach is not a very demanding game. Probably because it was built and optimized for the Xbox 360's GPU, which was released in 2005! 2D games like Age of Empires are much less demanding than Halo Reach too.
I'm not sure whether your $150 budget is just for a GPU or an entire upgrade; I have to assume just the GPU because an entire upgrade would cost much more.
If you're going to buy a new CPU:
The AMD Ryzen 5 3400G CPU has integrated graphics.
It can achieve 53 to 69 FPS at 1080p on Halo Reach.
Technically buying this CPU means you'd be spending $0 on a GPU if you accounted for buying a CPU anyways.
If you are just buying a GPU:
Consider the NVIDIA GTX 1050, 1050 Ti, or the AMD RX 560, RX 570. These four cards are around the $150 - $250 price range. The RX 570 is particularly good value (FPS per $) if you can catch it on sale.
If you just want to run Age of Empires for as cheap as possible, practically any integrated graphics should be able to handle it. It's a remaster of a very old game, and the higher resolution does not add much work for the GPU. So if you have any video output at all on a PC, it's probably good to go at $0 cost to you.