Can some ancient astrological architecture (such as stone hinge) be dated based on where the stars would have

where the stars would have been at the time? Certainly, since the time they were built, the earth has shifted. Therefore, based on how long it would have taken for them to go from the optimum positioning to where they are now would tell us when they were built (or finished), correct?

Yes. An Australian government astronomer named George Dodwell observed over a hundred astronomical records from various sites including Stonehenge, Egypt, China, India, Peru, etc.. and concluded that the earth's axis tilt must have been altered starting around 2345 BC and ending around 1850 AD. He could also calculate approximately when those markers or sites would line up with solar positions thus dating sites more correctly.

Dodwell's observations...
http://www.setterfield.org/Dodwell_Manus…

Related link with more info on Dowell and his measurements...
http://www.abovetopsecret.com/forum/thre…

Another link...less info
http://www.science-frontiers.com/sf030/s…

Only if the layout of the stones were absolutely known, but much is speculation and many parts are missing. One story set in 1250 BC worked off of some archeological evidence of poles and said that the poles height marked the location of specific stars at specific times of year - but those poles are long gone and their heights unknown.
And the purpose of the those places which were astronomical/astrological was to SET the calendar - when the sun shined on a stone it was THEN the spring solstice. But we have no absolute record of when that was. We can't say it was 3 hours or 3 days sooner than now, which would be needed to do what you suggest.

There is no "ancient astrological architecture." There are megalithic structures like Stonehenge which possibly were used for astronomical purposes, but they were probably only used to observe the Sun.

Mostly no.
Stonehenge still lines up with the Sun at the solstices.
Stellar movements are too slow and our history too short.