A Galactic year is estimated to be around 250 million "terrestrial" years. That means there will be about 4 rotations per billion years. And age of the universe is estimated to be near 13.8 billion "terrestrial" years. It looks improbable to think that our galaxy has rotated 50 times since the beginning of time. Could someone explain this please?
Your calculation sounds correct. It is however based on assumptions that are non-trivial. An analogy with the Earth would give that the Earth rotated around the Sun 13.8 billion times since the Big Bang. Which is meaningless since the Earth was created only a few billion years ago. Our galaxy, the Milky-Way, may have had a long and quiet history since 10 billion years ago (http://arxiv.org/abs/astro-ph/0702585), which is not too far from the age of the Universe. Our Galaxy, however, evolved, and its rotation speed too. As for the solar system, the duration of a "year" (ie one rotation over the Sun) on Earth is very different from the duration of a "year" on other planets. It is the same in the Milky Way where the revolution period is different depending on the distance from the center of the galaxy. As an order of magnitude, and considering in first approximation that our Galaxy is old, and that its properties did not change too much, an order a magnitude of a few tenths revolutions since our Galaxy is a stable disk sounds plausible. As a first approximation. I hope this will help you to find that such a value is not improbable.
Currently Astronomy relates to Time in a manner similar to the Geocentric / Heliocentric syndrome that plagued early Astronomers that did not have enough information available to accurately describe how the Solar System maneuvered.
Earthlings use Time as it relates to Earth to measure the Age of Celestial objects because we don't currently understand how Time works. More specifically we use the J2000 epoch to tell time on Earth because it was designed hundreds of years ago, appears to work for the time being and is commonly agreed upon.
The problem is that we believe our planet is approximately 4.5 billion years old and a year is an approximate amount of time that is consistent over long periods of time. So we extrapolate out and use that to measure objects outside our system since that is the only reference point we know and understand and we do not have access to a Time system that explains other Systems and is capable of measuring them.
The Universe if it is the largest encapsulating structure needs its own time system that starts from its origin and is consistent to measure against other systems. For example using a second instead of a year (presuming a second takes the same duration in all systems).
A Galaxy would need its own Time reference from its origin with in the Universe using a similar consistent standard like a second so you could accurately compare their ages. If you just look at the Earth year we don't know how long a year was a billion years ago, it may have only been 100 days and we don't know how long one will be in a billion years, it may be 1000 days, just to make an example. So an Earth year is a poor way to measure the age of other Celestial objects but it is the best we have at the present time so we continue to use it. Technically we shouldn't be able to measure anything older than approximate 4.5 billions years because theoretically that time doesn't exist because Earth didn't exist. You can see how this resembles the Geocentric / Heliocentric problem.
In the future we will realize the J2000 epoch is based on the birth of Jesus Christ and that we are not accurately measuring time on our own planet based on its beginning but rather basing it on an individuals life. At some point we will agree on a common starting point for the planet and develop technology to determine that moment similar to how we developed carbon dating.
You can see how this will even effect the term light year as we realize the length of a Earth year changes over long periods of time and is not accurate to measure distances that light travels relating them to orbital period of our planet. Note also light appears to travel a linear fashion whereas our planetary orbit is circular. The key would be to have a time standard that is linear in fashion as well.
Your point about how many times the galaxy has rotated is similar to the age of an earth year problem, no one from Earth really knows how fast or how long a galaxy evolves so they are just estimating based on the best information available and that is commonly agreed upon.