For a star that got ejected at near light speed from the galaxy, any orbiting bodies would probably be flung in other directions and no longer orbit the star.
Assuming the planet miraculously hung around for the ride, after the initial acceleration, it would be no different then what we experience here on earth.
Remember that everything is relative and perspective is key. A question about speed depends on who you’re asking.
To us, the other star and it’s planets are traveling at near light speeds away from us. To that stars system, it’s Earth that’s traveling at near light speed away from them.
Space flight would be the same and so would gravity and time. To them, nothing changes within their stars system because it’s all within the same reference frame.
I guess the night sky would be the main difference for people on that planet. They’d see the galaxy they left behind as red shifted and anything they’re speeding towards as blue shifted.
Since they’ve left the galaxy and was red shifted, they probably wouldn’t see as many stars with their naked eyes.