How can I paste the clipboard into a text field with the remote?

What is the context ?  For entering Server info ?  If it's for server info, the best option is to use the QR code or IP method listed at top of server screen and use your phone/tablet to enter that info.  A hell of a lot easier than using a remote.

Now for other apps, the only way I know how, is to switch to mouse cursor mode (mouse pointer button to bottom right of DPAD).  Then move to the entry box/field, press OK to select it, and then you can long press the OK button as you would on a phone screen, to get the context menu which will often include options like "Select All" "Cut" "Copy" "Paste"...  Then you select which option you want and hit OK again.

I have not tried it on the Classic AI specifically, but so far on most buzz boxes, that works.