Will Planets in Very Bright Galaxies Have Black Skies?

For example if I live in IC 1101 will I see the sky bright all the time?

IC 1101 is the single largest galaxy that has ever been found in the observable universe. It is located almost a billion light-years away.

It depends on where, exactly, you are inside that (or any) galaxy.

The Milky Way is a huge, bright galaxy and our skies are full of tiny dots in an otherwise black background. Why?

Distance.

The closest star to us is more than 4 light years away. At that distance, it matters not how bright the star is - it will be a dot in our sky.

In order to have a sky that is “bright” with stars, we’ll need a lot more stars, and those rather close. There are very few galaxies that have that kind of density. In fact, the most likely place such a world could exist is not within a galaxy proper, but rather within a Globular cluster - Wikipedia.

As you can see from the link, the centers of some clusters have stars packed together like planets in our solar system. If we lived on a planet orbiting such a star, it would never be dark. Instead, the “night” sky would be full of numerous stars so close as to appear like suns, bathing our planet in perpetual daylight.

I bet you can see the problem here: such a world almost certainly cannot exist. Its orbit would be horribly unstable thanks to the hulking stars lurking around every orbit. What’s more, even if such a planet existed, it would be a very warm place, unlikely to have any of the nice liquid water almost certainly required for life.

So is there a real world with at least a somewhat bright nighttime sky? Apparently the answer is yes:

Proxima Centauri b - Wikipedia

This planet orbits a dim star (Proxima Centauri) with two nearby bright stars. These stars are close enough to Proxima b that the nighttime side would be considerably brighter than the nighttime side here on Earth. What’s more, you could almost certainly see these companion stars during the day.

In the end, “bright” is a bit of a vague term - our full moon makes our nighttime side rather bright. But the answer to your question as posed is no: planets that close to other stars are unlikely to remain in stable orbits and unlikely to be cool enough to exist for long.