I'm not sure if I understand correctly...
If you just ask: can I connect Nucleo to laptop via ethernet jack and check the Nucleo's webserver?
Then the answer is: yes!
I recently did that while travelling on a train
But you either need some PC app to run it as a DHCP server, or you have to give the Nucleo a static IP address, with most parts matching that of your PC's.
subnet mask and gateway must be identical, IP4 address itself has to match the first 3 (or 2?) bytes.
Example:
laptop's address (find out via cmd / ipconfig)
IP4 address: 169.254.111.7
SubNet: 255.255.0.0
GateWay: 169.254.111.1
then your Nucleo's address should be set to:
IP4 address: 169.254.111.x
SubNet: 255.255.0.0
GateWay: 169.254.111.1
with x any 8 bit value except the laptop's (7 here)
I just tested that (but with a USB-ethernet bridge adapter with a Microchip LAN9512), it's working here.
Edit:
It will probably 169.254.x.y
wikipedia says:
| 169.254.0.0/16 | 169.254.0.0–169.254.255.255 | 65536 | Subnet | Used for link-local addresses[10] between two hosts on a single link when no IP address is otherwise specified, such as would have normally been retrieved from a DHCP server. |