What you'd have to do is:
1) Forward ssh and http to your Slackware box (so that it deals with incoming connection for these protocols); there should be something in your router's config to handle this.
2) Set up your ssh and web servers.
3) Sign up for something like dyndns, which associates whatever your current IP address is with a URL like http://javamdk.dyndns.org
. Dyndns is free for home use I believe.
That should be it.