How do I open "apt" links in Firefox (in Lubuntu)?

Point the select application to:

/usr/bin/apturl

This sould solve the problem.

Don't forget to check "Remember".

All you need to do is install the (Ubuntu) Software Center in Lubuntu.

Open a terminal, and type sudo apt-get install software-center. Then, the first time you click on a APT link in Firefox should give you the below popup, after which such links should open automatically:

Beside Ubuntu Software Center, a much lighter application (AptUrl) can be used to handle "apt://" protocol/links: see this answer to a related question.

The issue is solved after installing Ubuntu Software Center, which was not installed.

sudo apt-get install software-center

Now, after opening that link in Firefox, it offers the normal option to open that link in Ubuntu Software Center, and to remember that option. Also, following this, apt appears in the list from Firefox' Preferences/Applications:

Here is what Firefox support page says about this:

The Applications panel has limited functionality for editing. You can change the action for an existing file type but you cannot add or remove file types. Entries are added automatically when you download files and select actions for them.

However, in your case you can try modifying you ~/.mozilla/firefox/\*\*\*\*.default/mimeTypes.rdf. Here is a link to mine: http://pastebin.com/4HhX6xSE