The speed the new (and old) HP Switch shows on the port is the auto-negotiated one with the device it sees connected on the other side of that "network drop": so if there is an Unmanaged Gigabit Switch on the other side - on that network drop - the expected speed should be 1 Gbps if both Switches are Gigabit Ethernet capable (the old HP 4000 is not if I recall correctly).
This, obiouvsly, if all involved ports - of the Managed Switch - aren't set with a fixed Speed settings (which is non default).
The PC connected (via a Patch Cord few meters long, I presume) to the Unmanaged Gigabit Switch will probably always show you the 1 Gbps port speed. That's expected.
That's also happens if you connect that PC directly on the new HP Switch.
I hope to haven't misunderstood your example so here the question: What will happen if you connect your PC (with its 1 Gbps interface) to the "network drop" instead of directly to the new HP Switch's port?
Does the result change?
This practical test is to understand if the cable (directly connecting both the network drops <- to be verified!) is (or could be) partially/completely responsible of any sort of signal degradation.
if:
- involved Cable (and Patch Cords) is Cat. 6 certified (better than Cat.5E certified),
- involved Cable is shorter than (let's say) 90 m,
- PC/Unmanaged Switch *and* the new HP Switch both have all related port's speed set to "Auto" (e.g. Auto 10/100/1000),
then the auto-negotiated speed should be the fastest both devices are able to advertise and agree each others (so 1 Gbps).