Doug Drexler - who designed the ship, and is probably the best source we have - on the Enterprise-J:
Note that the Enterprise-J is only a ship of the 26th Century. So the Federation is making transgalactic voyages by the 26th Century, and it's 5 centuries beyond that in this scenario. Also, site-to-site teleportation is considered a pretty economical way for moving about a ship. Big implications for industry there.Drexler saw the Enterprise-J "as a multi-generational vessel, that had large parks, entertainment zones, and entire universities on board. The ship is so large that turbolifts would be replaced with site to site transporters. [It] had one deflector, recognizably descended from the NX. I opted for spindly nacelle struts because I felt it suggested a technology beyond what we were familiar with. They are beyond transwarp. They can fold space, and they are exploring other galaxies besides the Milky Way." The doors shown in the corridors aboard the Enterprise-J appear to be the same used aboard Deep Space 9.
By the 25th Century, the Federation has reached the Delta Quadrant (so, very likely a galactic empire). By the 26th Century, the Federation has absorbed at least the Klingon Empire.
In the mid-twenties it appears as though Federation technology has the ability to destroy entire solar systems and halt nuclear fusion in stars. Whether they've weaponized any of this technology is unknown. By the 29th century, one technology which is clearly a standard armament is the temporal disruptor, of which a 'force-3' version will 'shatter spacetime' with a radius of 150 kilometers (it was going to be used to destroy Voyager, so it would appear as though shattering spacetime also shatters the stuff in it). It can also be planted 'out of phase' to prevent it from being disarmed by conventional means. The only one that has been seen was planted on a ship rather than being a torpedo or anything, but given that by this period they have temporal transporters that can teleport something virtually anywhere in space and time - and through conventional shields, as well, showing none of the finickiness or any evidence of any of the weaknesses of regular transporters - this isn't such a huge problem. I believe we have our Deathstar-killer. The only other piece of evidence we have for firepower from Star Trek's future that I'm aware of are the 24th-Century transphasic torpedos that can one-shot a Borg Cube. Oh, and I guess Red Matter, if they're into that sort of thing. But there's nothing concrete for the several centuries thereafter.
Time travel by the 31st Century is also "so commonplace that there are quantum discriminators in every high school desk."
Frankly, we don't know much. But from the vague sketch we have, even several hundred years before this match-up they're a transgalactic time-travelling magictech space empire. I think there's good reason to think they could take this, simply because I doubt the Empire has any defense at all against many of their technologies.