Don't do your idea!
See Is an 'if password == XXXXXXX' enough for minimum security? for some basic password advice for applications. In virtually all cases, a password should be hashed so that the value cannot be retrieved.
Important: If you do continue with your idea, you've just stored retrievable passwords for several applications in a single place, which is much, much worse than violating the retrievability principle for one application. So, even if those applications had implemented good security practices, this interception step breaks all of them.
Single sign-on is usually implemented through some method of either sharing an authentication token (could be used where each application is produced by the same vendor, sharing similar database fields and being able to authenticate the validity of that token), or a single authentication server (OpenId) or one of the other means listed in that article.
It's always very complicated and requires very advanced understanding of this topic and security in general.
So, your first impression was correct - if SSO is involved, there is no "easy" way to do it. The SSO requirement makes this project sound like it's beyond your current means, especially if you were asked to "create a web page" that provides SSO.
- A single web page with links to different applications is obviously trivial, and I suggest you complete that as a first step.
- Next, I'd suggest you research SSO, understand how it relates to your applications, and take this back to your manager.
- If your manager would like to continue, I would look at existing SSO products. Building a secure in-house solution is probably more than you want to take on. If you do decide to build it in-house, get learning.