Yes. To see this, use two people instead of one, and have the second person walk down the hill while the first person walks up. They obviously must meet at some point, and if the second person is following the first person's future trip down, then where they meet is the point where the first person is at the same place and time on both trips.
But if "time of day" changes during the trip, then the answer can be no. This is because the time of day on the second leg could change just as the person is crossing the intersection point.
Time jumps ahead by 1 second worldwide every year or two when the International Earth Rotation Service determines that a leap second is needed. See <http://tycho.usno.navy.mil/leapsec.html> and related pages for further information.
So the answer is no in general, but yes if it is restricted to places where a leap second could not occur between 8 pm and midnight. Now, the only time when leap seconds are inserted is at the end of certain months in Coordinated Universal Time (UTC; see the above web site for details). Hence, for those locations the answer to the puzzle is yes.