Internet bridge from house to shop.

So I have fiber to my home and router in my home.
Can I not run a cat5 from that router to another router in my shop?
Yes.

Routing the cable may be the hard part. At 100 feet you will get almost 700 Mbps transfer speed between the two routers. And most Cat5 cable is not Direct Bury and will rot out.

If the two routers use different SSID's you will not be able to print anything from the shop to a Printer in the house. They must both be the same LAN.
A lot of new routers have 'MESH' where they all, up to 6 I think, have the same SSID and auto switch to the closest one if you move the laptop around.