Your iPhone can work with any Apple Watch starting from the original one to the latest one. The only thing that needs to be ensured is that both the iPhone and Apple Watch are running the compatible versions of iOS and watchOS.
It's recommended that the devices run the latest available version of the respective OS. Currently the latest version of iOS available for iPhone 8 Plus is iOS 13.2, while the one for Apple Watch Series 4 is watchOS 6.0.1.
To answer your questions specifically:
Can I use my Apple Watch 4, having only an iPhone 8 Plus?
If not, what is the latest Apple Watch that will pair with my phone?
As stated above, you can pair technically pair any Apple Watch with iPhone 8 Plus.
Can I get my Apple Watch working without buying a newer iPhone?
If not, what is the latest generation Apple Watch that I can use without buying a newer iPhone?
Apple Watch Series 5, which is the latest model released by Apple.