Help with my Omega Pocket Watch

Welcome Big Mamma E. The watch is made in Switzerland, but could have been purchased anywhere. If I had to guess an age I would say it is about 1920, see this similar example
https://www.antiques-atlas.com/antique/omega_steel_open_face__pocket_watch_swiss_1920_a236/as834a236:



As @ghce says, you can get a more accurate date of production form the movement serial number. I agree that $500 is too much for the starting cost of a service. I would expect to pay more like $300 US, but costs vary by country. The cost of service will likely exceed the resale value of the watch, so only get it serviced if you plan to use it. I suggest that you get a nice way to display it alongside a photo of the person you inherited it from. I had a nice case made for mine. If you send me a private message, I can share with you the contact info for the guy who made it.

The case serial number suggests maybe 1916-17, but having both the movement and case serial numbers is always helpful. You are definitely going to need a new hour hand, and probably easiest just to get a hour/minute matching set. A watchmaker could help you with that.

You could shop around and find a cheaper service, but since this is a one-off job for you, I would think finding a competent and knowledgeable watchmaker would be more important than price. And more expensive doesn't always mean better. I would suggest gathering some firsthand recommendations, and don't be afraid to ship the watch to a good person.