Can you download the puzzle and answer files to your computer for printing out later? (E.g. by right clicking a link and choosing Save as... or by going to the File menu of the browser and doing that?) If so, you could (assuming no other method is proposed) try something like this:
Put all the puzzle and answer files in the same folder. Using Windows Explorer...
Easiest if you choose Details or List view, maybe?
Rename or prefix the names of the puzzle and answer document files so they sort puzzle-answer-puzzle-answer e.g.
01 Puzzle 1.doc (or pdf or whatever)
02 Answers 1.doc
03 Puzzle 2.doc
04 Answers 2.doc
05 Puzzle 3.doc
06 Answers 3.doc
(etc)
Sort the files by name so they appear as above.
Hold down the Ctrl key and click each odd numbered file (starting 01, 03, 05 etc) in turn to highlight all the "puzzle" files. Right click any one of them and select "Print". When they are all done take the whole stack out of the printer and compose the stack so that the back of puzzle 1 is on top, followed by the back of puzzle 2, etc. Put the whole lot back in the paper input tray the right way around and now highlight all the even numbered files (starting 02, 04, 06 etc - the answers) and again right click any one and select "print". At least this way you are not having to reverse each sheet at the time of printing it.
Or if you have to do it from the browser you could print out a stack of puzzles and then turn them over and go back and print the answers to those puzzles in order