Good one. To begin with, assume no rest breaks & constant speed for each journey (simplified version) Draw a line segment on a page, label one end A the other B [A is the botton of the hill B the top] Above the line,mark the monks position at zero point, Day one, 12 hours Day one 24 day one, etc. using 24 hour clock. BELOW The line do the same for the return journey. As you can see, the times increase Left to Right above the line, and Right to left. Times actually cross at two points:
approx 04.40 Day TWO outward, and 04.40 Day 2 Back.
19.30 Day TWO Outward,and 19.30 Day 2 Back.

With rests, sleep etc, we cannot calculate exactly WHEN he will cross, (Lets take a ridiculously extreme example, where the hill is really short. Up hill he walks five minutes, rests almost 3 days, walks five minutes end! Return journey he rests for one day less!). However, if you draw the diagram for the simplified version above, it is clear that he MUST cross. Normally, he must cross TWICE, however, you can bring it down to crossing only once, if he stays in one place for a full day.