I would like to remotely login to a server on the intranet of our company for some system maintenance work during the holidays: our mailboxes should be cleaned otherwise the mail partition will be possibly filled up with spam.
I do not know how to connect to that server and all the information I have is as follows:

- the route I have to pass through is possibly as follows:
internet -- our ISP -- firewall on our intranet -- intranet server

- our intranet is a private network, not having a real IP address, only a dynamic one

- our servers are always on

- I know the administrator of our ISP can remotely login to any of our servers, therefore I suppose that SSH may be listening on one of the ports on the firewall of our intranet (maybe port 22?)

- some users can reach their mailboxes located on our intranet server from the internet

- I know the root password on the server to which I would like to connect on the intranet

- I also know my username and password, besides my RSA authentication key is in the authorized_keys file in my home directory of that server. I also know the cipher type to use.

- ping 'servername.some.where' gives me an unknown host error - ping 'www.some.where' gives me a result, but that's only our web page located at our ISP

So, is there a possibility to login to our intranet server passing through our ISP and our intranet firewall based on the above information?

The essential information is if you can ping your firewall. If you can, you can also login into your server using ssh (as your firewall seems to be allowing/passing ssh connection).
There can be 2 hard points.
One: your ISP is not passing ssh requests to your firewall.
Two: your firewall is a hardware one, not allowing to log into it (and make another ssh connection to one of your servers) and your firewall is not configured to forward ssh connections into one of your servers OR your firewall has a local IP.

I do not know how to ping our firewall.
As I mentioned before I pinged www.some.where and servername.some.where with different results, but I suppose these pings did not get through our ISP.
Previously I found an IP address using netstat on that server - I thought it was our actual IP address. Now I do not think so: looking up that IP address gave mail.some.where (which is possibly at our ISP). I also tried tracing www.some.where which gave me the same IP address. (It may mean something or nothing - I do not know).
I am convinced that it is possible to get through our firewall, since our ISP does it often. I am not sure, however, if our ISP passes through SSH requests to our firewall. I could find it out if I knew how to ping our firewall.
So, how to do that?

Try to ping your firewall. Or better: run traceroute to any host in the Internet (may be linuxquestions.org). You'll get a list of host your packets are passing by. The first one should be your firewall. Look as many local IPs there are.
It may be that your firewall has local IPs, the same with most of your ISP's network. In such case, they can use ssh connection to your server and it's hard to do it from outside.