A quite direct argument is to solve exercise 4.4.17 in page 252 of Thurston's 
book 3-D Geom & Topo - the published one in 1997. When doing this exercise, 
one may want to notice two things: 1. On $S^2$, each of the $5$ angles of a 
spherical pentagon face of a dodecahedron is $120$ degrees, in contrast to 
$108$ degrees for a pentagon on plane - this is useful when you try to create a local coordinate system on the circle bundle $UTS^2$; 2. The 
"fundamental domain" Thurston created here does not look like a dodecahedron 
in the metric sense - it look like a cylinder over a pentagon base; however 
other "fundamental domains" will mark a line (with quite a slope!) on each 
of the five side faces and therefore each side face splits into two "pentagons"... 
With these in mind, exercise 4.4.17 should be very pleasant to work out.