I think the oldest cartridge that would still work in a modern firearm would be .22 Short, which is from 1857.[1] It was developed into .22 Long, which was developed into .22 Long Rifle, which new firearms are still chambered in today. It wouldn't cycle a semi-auto though. Another contender would be .38 Long Colt, which is from 1875 and will work in firearms chambered in .38 Special or .357 Magnum.

The main causes of incompatibility were the transition from ball-and-powder to rimfire cartridges, then the transition from rimfire to centerfire. It's surprising how little firearms have changed in the past century. 9mm Parabellum is 125 years old. The Browning short recoil action (used by most handguns today) is over a century old.

1. https://en.wikipedia.org/wiki/.22_short

If you live in a country or region where the gauge didn't change, you can put a 1845 train on a modern track just fine. If the gauge did change, you'll need a train from another country. Standard Gauge was first applied in 1825 after all.

Many militaries are still buying rifles that chamber a cartridge introduced in 1891 (7.62x54mmR). I'd argue a cartridge initially developed for a long bolt action rifle (Mosin-Nagant) that can now be fired out of a variant of a AK-patten select-fire bull-pup (Vepr) fits the analogy.

>whether you could put a train from 1845 on a modern track

you can do that, in fact I know that Austria still has a mid 19th century train in regular operation (https://de.wikipedia.org/wiki/SB_alt_23)

if you're looking for commuter trains a lot of early 20th century ones still operate around the world.