First, to make one thing clear: when you visit a web page like http://example.com, your web browser is actually making a request to example.com:80. This is done transparently because port 80 is the standard port for the HTTP protocol. As you know, you can request a non-standard port by appending it to the domain name in the URL: http://example.com:888/.
Unfortunately, you cannot have a domain name "alias" that somehow includes a non-standard port - your browser will always try to use port 80 if you don't specify a port.
One solution would be to use a proxy - nginx, apache, lighttpd, and others can all do this.
The idea is that you set up a proxy server that is listening on port 80 on your host. It waits for connections, then forwards those connections to a different server (on the same host, or on a different one) based on some rule. So, for example, you might have rules that look something like this:
IF host = teamcity.companyname.co.uk THEN forward to teamcity:9090
IF host = octopus.companyname.co.uk THEN forward to octopus:8090
The syntax for these rules vary widely between different proxy configurations, so this is just an example.
Note that this is not a redirect - the user's browser connects to teamcity.companyname.co.uk for all requests. It's the proxy that sends the request on to a different service and forwards any responses back to the client "behind the scenes".
These proxy configurations can get quite complex. For example, what if your teamcity application serves a page with a link on it that points to http://teamcity:9090/path/to/page? The user's browser is going to fail if they click on that link. Fortunately, proxies can be configured to rewrite URLs like this on the fly. You'll need to do some research to tailor this solution to your situation.
