Hello! As far as I know, you can't really change the resolution of your printed tree, only configure how large you want the tree to be (i. e. on how many pages you want to have it printed on). When you are in the print menu, you an always check how the tree is going to look like on paper via "Print Preview" or"Downloa PDF" (though the quality might be influenced by your printer's quality as well).
If an A4 paper results in illegible names because of font size, maybe try printing on a bigger format (that you can choose in the Print menu)? You can usually do so in copy shops or order from the internet.
Another possible solution might be exporting your family as GEDCOM (via the "Save" button on the bottom of your screen), uploading it on another geneaology site and trying their option to print? I have no idea whether this will work as I have never tried printing my family trees, so this is just idle thinking from my site.
