Unfortunately, I don't think there's any particularly easy way to find a specific problem given its index number, but let me summarize some of the comments (and add some of my own) in a community wiki answer.
The American Mathematical Monthly is published electronically on the Taylor & Francis website. You can often get a free first-page preview of the Problems and Solutions section, so one approach is to manually step through each issue and click on the preview. For example, you mentioned Problem 12412; using this manual method, I found it in Problems and Solutions, Volume 130, Issue 8. In this case, the preview page actually provides the full text of the problem in question.
Even if this method doesn't yield the full text of the problem, it should let you narrow down which issue the problem appears in. After that, it's a matter of tracking down the actual content. Options include:
- Becoming a member of the Mathematical Association of America. I don't know what your status is, but for example, a student membership is $35/year.
- Going to your local college or university library. I don't know how convenient this is for you, but this will likely allow you to get access to the content for free.
- Finding someone who is willing to share the content with you. This might be a math professor at the aforementioned university, or someone you can find online. Here is one website which invites people to share.
For content more than three years old, you should be able to use JSTOR, but it's rare that a problem that old will still be "live."