Get date and time by position of the sun and the observer position
I think you can very easily use the transformations available in the wiki link to obtain the Julian day, then exploit the position of observer to get local sidereal time.
Lets start by computing the Julian day
you have a given longitude $L$ which gives you $n$ number of days since Greenwich noon as
$$n = \dfrac{L-280.4860}{0.9856474}$$
giving the corresponding Julian day as
$$JD = n+2451545$$
once you have Julian day you can use the following link to obtain UT
http://ssd.jpl.nasa.gov/tc.cgi