As I understand it you have a photo something like this:

This is landscape orientation.
If you want portrait orientation (with the depth greater than the width) you can either (a) crop, which will remove some of the subject; (b) squeeze (which will distort); or (c ) use image editing software to resize the document canvas and then fill the extra space with more background:

None of these is going to be particularly effective. But depending on your printer, and the software you're using to open the image, you should be able to fill the paper.
These are my print options in Photos:

It does a pretty good job of fitting the image to the paper, although you do have to physically rotate the print by 90 degrees after it's been printed 😉