Real-time position of a distant celestial body How might it be possible to determine (via some specific online empemeris or via some well-understood algorithm) the observed position of some distant celestial body (e.g. Betelgeuse)?

To be concrete: suppose I wished to do what any number of mobile phone applications do, i.e. computationally model the observed position of stars and project them onto the screen using GPS coordinates and compass direction, so that an `overlay' of the night sky is produced. Then:

a) Where could I source the relevant data from?

b) What coordinate system would it be best to use?
The most robust, but generally not computationally feasible way would be a full N-body simulation of all the relevant interacting neighbours of the object in question. 

On a more crude note, one could observe the eigen motion of the object (its velocity vector), its distance to us, and from that information (assuming the eigen motion is constant over the period between now and when the light was emitted from the object) one may infer the current position. 

You can of course make a hybrid of the above methods: use the observed motion of the object, and use simulations to include the interaction with other objects.

Note that 'current position' is not unambiguously defined. Astronomers would often consider the observed position to be its current position, even though we are looking back in time. And, needless to say, on cosmological scales these concepts pose more problems.