My #1 network troubleshooting tool is called PortQry. Microsoft provides it as a download here in command line version and here with a GUI front-end.
This is how you would find out if a machine had remote desktop running:
C:\>portqry -n server01 -e 3389
Querying target system called:
server01
Attempting to resolve name to IP address…
Name resolved to 10.1.1.1
querying…
TCP port 3389 (unknown service): LISTENING
Since ICMP (ping) is disabled so often these days PortQry is now the new ping.