Can I buy an Apple Watch and set up the monthly payments for the Services on my Card?

No, I doubt that you can do that. The watch's data plan is an additional add-on to the iPhone's existing service plan.

Perhaps his employer will make an arrangement for him to add the data plan and pay for it himself. Perhaps a paycheck deduction. The plans are $10 to $15 a month extra. But the service provider must support Watch.

However, this article describes using a gps/cellular Watch with a family setup that doesn't require an iPhone, but has a few less bells & whistles.

https://www.makeuseof.com/family-setup-apple-watch-without-iphone/

M1 Mac mini, macOS 11.3.1; Watch, series 5 GPS/cellular, watchOS 7.4.1; iPad Pro 2nd Gen, iPadOS 14.5.1; iPad Air 2, iPadOS 14.5.1; iPhone 12 Pro, iOS 14.5.1; TV 4th Gen, tvOS 14.5  HomePods, audioOS 14.5 and Airport Express w/ Airplay 2

You would have to check with his company.

Some corporate cellular carriers plans DO NOT allow "Split Billing". Some do (AT&T does). HOWEVER, the company must agree to allowing Split Billing.

So need to talk to his company.

The alternative, which is much more expensive, is if the company iPhone has Dual SIM support (physical SIM and an eSIM), you he could get a full price private cellular plan for the eSIM and pair the Apple Watch to the private plan. This means he could ONLY answer private plan phone calls on his Apple Watch, as well as private plan SMS/MMS messages via the Apple Watch. iMessage and other services that are strictly Internet based, would not care.

NOTE: The GPS only Apple Watch can do Everything the Cellular model can do AS LONG AS your iPhone is nearby, where nearby is generally somewhere in the same moderately sized home. The Cellular model allows you to leave your iPhone at home while you go for a walk, run, bike ride.

First, you should make sure his company allows a watch to be paired with the company iPhone.

Especially if it is a cellular watch and he plans to use that feature. You need to make sure that you have an eligible cellular service plan with a supported carrier. Your iPhone and Apple Watch must use the same carrier, and you must be within your carrier's network when you set up cellular on the watch. For more info, see:

Set up cellular on Apple Watch - Apple Support

As far as payment options that depends only on where you buy it and what their payment options are with your card.