[paul.fr]

the earth orbits the sun, so does the sun go in orbit around the universe? if yes, how long does it take?

[DoctorBeaver]

The galaxy that we are a part of is rotating.

The Sun is orbiting with a speed v = 220 km/sec = 0.000225 parsecs/year.
The radius of the Sun's orbit around the galactic centre is a = 8000 parsecs.
The circumference of the Sun's orbit is then 2 pi a = 50,300 parsecs.
The orbital period of the Sun thus turns out to be P = 2 pi a / v = (50,300)/(0.000225) = 220,000,000 years.

It takes the sun 220 million years to circle once around the center of our galaxy. THe sun is approximately 4.6 billion years old which means that it has orbited the galaxy just over 20 times.

[Soul Surfer]

To take this matter further our galaxy is moving as a whole with respect to the cosmic microwave background but this does not appear to be an orbit around anything.  It is moving and accelerating towards the Andromeda galaxy by mutual attraction and also has a random component of velocity  but there is no definable centre associated with the whole of the visible universe about which anything is likely to orbit

[DoctorBeaver]

If the whole universe were revolving, how would we know? There would be nothing to measure the rotation against.

It would be like living on the inner surface of a hollow sphere that was rotating. You and everything that you could see would be rotating about a common centre, but there is no way you could ever know it.

[lyner]

Not necessarily, my friend. Coriolis force would tell you if you were rotating. You  would need some sensitive gear to detect it, though.

[Ophiolite]

..... there is no definable centre associated with the whole of the visible universe about which anything is likely to orbit

This is something of a nipicking point, but I find it useful in explaining why there is no definable centre. Of course, there is a definable centre: we are at it. We are the centre of the Universe, of course, so is everything else. If the Big Bang is valid, then everything, starting as a singularity, was all at the same point - all at the centre. After expansion everything must remain the centre, though the word now becomes somewhat meaningless in this context.