I'm wondering how can I curve the text below to fit the curve of the shape below. I am somewhat new to Illustrator and yes I know my version is outdated
There are several ways to curve text in Illustrator, but the easiest is to select your text (It does not need to be outlined) and go to Effect -> Warp -> Arc... in the main menu.
The harder way (Not really that hard) is to draw an oval and use the Text on a Path tool to add text onto the ovals shape. The benefit of this method is that your text curves around the shape without being warped in any way.
Looking at the image above I think you should be able to use the 'make envelope' tool and adjust the curvature settings to match the object.
If you looking for more precision, you could use the meshwrap tool.
You could also try to create a curve that matches you objects curvature and type along the path.