I'm not familiar with paint shop pro but even so, the procedure will be the same or similar.
To print it well at 6x4 you'd need to have the image at 300dpi. in other words, 1800 x 1200 on your screen. That's fairly easy to do by increasing the image size and increasing the number of dots per inch. The big problem is that your original is very small. Increasing it to such a size will probably mean a large loss of quality. However since it's only a 6x4 adn should only cost a few pence or cents there probably isn't anything to lose.
I'd suggest a couple of things though.....
increase your image in a series of small steps rather than one single increase. The program you use can work out more accurately the extra pixels required if it has to insert less when increasing. So a few smaller steps should get you a better quality final image than one big step.
Also what about using your program to give a frame or border round the image to help frame it. It would also lessen the amount if increase from the small original and might look ok. I know it's small at 6x4 but if it's sentimental why not put some text in the border like the subject's name or some other personal detail?
I think you might be struggling to get a nice final image but like i said I'm not familiar with paint shop pro. I think Photoshop would struggle though.