Yes this will work, however you need to turn DHCP off on the old router or the two are going to conflict in giving out IP addresses. The IP of the second router may also need to be amended if it’s similar to the existing one; the default IP of the Virgin hubs is 192.168.0.1 I think, so just re-IP the second to be 192.168.0.2 (as long as you don’t have anything else on that address) and you’re away. If you aren’t comfortable doing that, buying a simple switch will give you the same effect but without WiFi.
Some other things to consider, you’ll want external grade cable if you want it to last, internal is cheaper but not suitable for outside. The maximum total length of 1 cat5e run should be 100m for best performance, as long as you keep it below that total it should be fine. If the man cave is 75 metres away from the house that’ll give you 25 to play with, which sounds a lot until you start weaving it around walls and doors etc
That’s probably the cheapest and easiest way of getting internet down there, however if there’s decent line of site a wireless bridge would also work. That consists of 1 access point on the house and one on the man cave linked together, which then gives you a cabled connection in the man cave. This takes more networking knowledge and is going to be more costly but misses out the running 75m of cable outside.
