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.
Do you already have the 150 to 175 foot piece of Cat5?

If the shop is 100 feet from the shop, add in the distance from the house router and where the shops router is, adding any ups, down walls and any bends. A 100 foot as the crow flies will end up much farther when you pull the wire.

If not spend the extra money and get direct bury Cat6. It has much better speeds at this distance. It also has a thin metal "DRAIN" wire that you can BOND at your Fiber Modem's Coax so if lightning does hit most of the spike will bleed out to the outside demarcation point.

When stapling this cable, do not dimple the sheath with the staples as this WILL slow down the bandwidth. The staples must be loose enough to move the cable back and forth. Never kink it. Keep all bends at a 6 inch diameter or MORE.

Arrow makes a T25A stapler for Cat5 and Cat 6 cable that does not dimple the sheath. The old T25 stapler will dimple it.