Finding a location using the direction of shadow, UT and date approximately
Planetarium software, such as Stellarium, or astronomical computation software such as pyephem can compute the exact position of the sun at any date, time and location. It is then simple trigonometry to calculate the length of a shadow:

if the sun is $\theta$ degrees above the horizon, a vertical building of height $h$ will cast a shadow on horizontal ground of length $\frac{h}{\tan(\theta)}$.