if you're sending alerts to jon.doe@hotmail.com for example, set up an SMTP server and make it able to relay. Then, add an entry in the hosts file on the machine the Alert Manager is installed. The entry should pint to the machine on which the SMTP server is running. The hosts file should look like this:
10.0.0.15 hotmail.com
where 10.0.0.15 is the IP number of the SMTP server.
By this the Alert manager will send the message to the SMTP server which will relay the message to hotmail.com. PLease note it is importnat for the SMTP server to have Internet Access (at leat SMTP access to the Internet!).