A person walks up a hill starting at 8 pm and arrives at the top at midnight.

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.

Mark Brader           | "But how do you figure out whether the programmer
Toronto               |  knew what he was doing when you find his code
msbrader@interlog.com |  after he's gone?"  -- Roger Critchlow