|I guess I'm misunderstanding you and what you're trying to achieve.|
Leaving the Linux box as a DHCP client and having to then check in the router's management interface to see what the Linux box's IP address is every time you wish to connect via putty makes no sense to me.
Guapo said it, I've said it....statically assign an IP to the Linux box so that it never changes. Problem solved. No need to go into the router's management console, no need for one in Windows.
Just FYI, when you asked, "f there is in the Windows (7 in the case) an "admin console" working in the same way as the router's one. In fact is a Windows 7 question, just that." the simple answer is "No, there is no such console" period, full stop.
If you're going to insist on doing things the hardest possible way you can find, here's a tip for you.....in order to find out what the Linux box's IP address is, it's a simple matter to open a Terminal Window and type the following command:
The output is your TCP/IP settings and will show the IP address the Linux box has been assigned by DHCP. Since your two units, whether in an ad-hoc WLAN, or connected to your router's WLAN will be close enough to each other to easily and quickly walk from one to the other, you can find the IP address using this method and then walk over to your Windows box and enter that IP into Putty and then connect.
But for the sake of clarity I will reiterate......this is convoluted and unnecessary if you simply assign a static IP and then save that info into Putty.