If it is a high-quality faucet there are usually rebuild kits for them. If not, the plumber's estimate sounds about right.

Nonsense!

Buy a new handle. Do you have the old one that broke or remember what it looked like? Can you see any indication of the faucet brand stamped on the hose bib?

It looks like you need a handle that looks like this:

(Repair kit for a Prier faucet; you can buy just the handle separately.)

Hose bibs are cheap, go to your local plumbing supply house with the pipe size.

They are very easy to replace if you have access to the other side.

Just get a new frost proof hose bib and ask someone in plumbing supply to get you quick connectors. and they will give you everything you need. Cut the pipe and put everything together. (don't forget to shut the water off)

Ask anyone you know a bit handy and they can do the swap in 10 min.

That said if you don't know what you doing or know anyone who might know... The price you got from the plumber is about right for the labor and materials.

Hi. Yes I have access inside Just wanted to go the easier route by adding an adapter handle or if I’m able to take the screw off (it seems pretty rusted) and replace the handle

In that case, spray rust cleaner on it and go over with a steel brush and spray some penetrating oil to loosen up the screw or any other rusted parts and replace what needs to be replaced.

Just had a plumber replace two broken frost proof exterior spigots: Materials (Two spigots, copper fittings, tube, etc.): $98.27. Labor: $170. Service call: $85. Total for two spigots: $353.27.