How to wire two offices on two separate floors together

If you're able, just bring all the wiring back to a central location (a "main distribution frame", or MDF). The typical reason you wouldn't be able to do this would be because of distance limitations between the MDF and the client. It sounds like that might not be a factor for you. You might also have a concern that the building could be split up into different suites later and you might want separate wiring closets ("independent distribution frames", or IDFs) to that end.

If you don't have a reason to split the wiring up into multiple cabinets, though, don't do it! That's just adding complexity. You're also potentially adding a bottleneck between the IDF and the rest of the network. You also have to consider physical security, power protection, and cooling in all the IDFs, too, potentially. A single switch or "stack" of switches in one location is typically easier to deal with in the long run (single locked door/cabinet, UPS, etc). Keeping the traffic in a single switch or switch stack is the least likely way to create bottlenecks, as well.

Typically you would not just patch the copper runs from the IDF-to-client into an IDF-to-MDF copper run. If you can do that then that implies the run length from the client-to-MDF is below the maximum for your networking technology (typically 100 meters, since most people are using Ethernet today) and there's no reason to even have the IDF!

If you do have to have IDFs because of distance limitations between the client and the MDF then the typical methodology is to run a group of "trunk" cables (either copper or fiber) between the IDF and the MDF. The switch in the IDF (typically called an "edge switch") connects to the wiring runs to the clients and one or more ports on that switch connect back to the MDF (typically called "uplinks"). It may make sense, depending on the bandwidth needs for the clients in the IDF to communicate with resources at the MDF or in other IDFs, to "bond" multiple connections from the IDF to the MDF or to use a higher bandwidth uplink technology (like 10G Ethernet for uplinks to IDFs with 10/100/1000 edge switches).

Sometimes IDFs are used because of a perceived savings on cabling expense (by using less physical wire). These savings need to be offset by the increased complexity of having a switch in the IDF and the potential bottleneck of the uplink port in the IDF becoming saturated with traffic from clients communicating outside the IDF.

If the second switch is unmanaged, then you're in for a world of pain. Without management features like Spanning Tree Protocol (STP), then you'll probably create some kind of loop, and that will invariably, eventually lead to a broadcast storm.

This is why if you're connecting switches together, especially to improve interconnect speed, then you need to configure LACP on both switches, so that they see it as a trunk path between the two.

I'm willing to wager that by connecting the two switches together in a naive way, then you'll either reduce network performance by encouraging routing loops, or cause broadcast storms which will bugger both switches equally.

Sell the unmanaged one, get an identical one to the first, and do it properly.