Check my 4 Apple e-mails on other than my own Mac?

I'm a little confused about what you're asking. However, let's see if I can help...

If you're wanting to check your email from any computer, you can do so in a browser: log into icloud.com with an Apple e-mail ID. Then, click the Mail item in the window.

You can do this from any browser on any computer, so say if you're in a library, Internet cafe, or borrowing a friend's computer. Be aware that if you do this on another Mac, it'll attempt to use the currently logged-in AppleID; however, you can click the link to 'Use a different AppleID' to use any of your own IDs.

If you're wanting to use a different e-mail app on a different computer, say, you can set up IMAP mail with your Apple email accounts. Different email apps have different setup requirements; however, these Apple Support instructions give you the settings you need for any email app that allows IMAP.

In this case, you'd have to find the instructions for whichever app you were going to use.

Are all these Apple-hosted addresses like icloud.com, mac.com, or me.com? If so, you can log into icloud.com from any web-connected device and read the mails.

Beyond that, lisfolks is right: we need more details. What email services (domains)? Do you have web access without your Mac. like an Apple mobile device?

How do you now check mail? A mail client like Apple Mail, Outlook, or ThunderBird? Through a web page?