Although the general theoretical description of the Universe is given by Friedmann–Lemaître–Robertson–Walker metric, and although it allows the Universe to be both finite (closed) and infinite (opened), scientific observation has shown that the universe began a finite period ago (approximately 13.798 Billion years ago). There was a big-bang, inflation of the universe. All motion of stars and nebula are travelling away from each other from an apparent singularity, and this 'bang' left residual heat and still evident structure (to the universe).
Along the 'time axis' in the negative direction then, the universe is finite. Along the time axis in the positive direction it may be theoretically possible for the universe to age infinitely, but it has't yet, and at every point aging in that direction the universe will still be finite back to its origin. At no point in the future will the age of the universe be infinite, and it isn't currently.
So the age of the universe is currently and will only ever be finite. (Besides, David Hilbert proved that it is physically impossible to have an infinite change of events, so even as we approach infinity, the universe's age will be finite, perhaps very large, but still finite).
