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According to general relativity time runs slowly near a massive body. So, I have a question bothering me from some time, that if time for every body/object in universe runs at a different rate, then how can we determine the age of universe? i.e. 13.82 billion years?

Near a blackhole as time runs slowly, which implies that the age of universe would be much much less. In fact, ever body/object in universe would have different age of universe according to it. And if we really want to calculate universe present age, we should put an upper limit to its age, by determining the time passed in empty space from big bang, which is away from gravitational source (as time passes the fastest here) and a lower limit by the oldest and massive blackhole in universe, as time there would have passed the least (i.e) the age of universe from its point of view would be even less than 0.5 billion years. Am I right?

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You can define the age of the universe at any spacetime event defined as a place and time as the longest time measured along any timelike worldline stretching from the event back to the big bang.

There will worldlines stretching back to the big bang with shorter times because they pass near a blackhole or because they are the worldlines of fast moving objects in the universe, but there is always a maximum time that can be taken at the age at that endpoint (assuming that there are no closed timelike curves of course).

For a point near the blackhole the longest age would be measured along a worldline that had only come near the blackhole recently.

You can prove that the hypersurfaces of constant time according to this definition are everywhere space-like (easy exercise) so the definition makes good sense.

There is no minimum time for such worldlines (e.g. a nuetrino moving at nearly the speed of light would have measured a very small proper time since the beginning of the universe) so your idea of a minimum age does not make as much sense without further constraints on the worldlines allowed for the measurement.

In practice we can just assume a homogeneous model and measure the hubble constant or characteristics of the cosmic background, and that is good enough for most places. But it is useful to know that there is a well defined age for any event, irrespective of how well homogeneity applies.

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