Well, I saw someone asking this question somewhere, so this post !
On the main menu on Nokia 5800 Xpressmusic, you might have seen a section called ‘Download’, This section is provided by Nokia to share useful softwares which can be used in the host mobile phone.
Following is a snapshot of the Nokia 5800 main menu, this screenshot is taken by using ScreenSnap.

On exploring the download section (You would need an internet connection to use the ‘Download’. If your mobile service provider has provided this option, its fine, else use the wifi option on the phone to connect to your home wireless network), one can find useful downloads such as applications, images, videos etc.

Go to the application section (your phone will take few minutes to refresh content from internet, this is normal, leave the phone to refresh content), you can see something like below.

Install the Zip application as well as other ones you need. You can now open zip files on your Nokia 5800 Xpressmusic. When you try to open any zip file, this application will do the job of extracting and opening zip file contents in your Nokia 5800 tube !
[UPDATE]
The above post was made when Nokia OVI was not released. After Nokia OVI was released, what you could find under downloads was just the Nokia OVI application.
Do not panic, just download Nokia OVI application, you may be required to create a new OVI account if you do not have one. After installing the app, start the app and search for the Zip application mentioned above. Install the Zip application from OVI, and that’s it you should be able to open zip files in your Nokia 5800 Xpressmusic.
