If (BIG if) we some how figure out how to launch a mass the size of a star or planet into an orbit around two galaxies. You could then launch your starship into its orbit around the apogee, ride the mass to the next galaxy, and de-orbit burn around perigee! In this way the orbit would practically be a store of energy as well.
The Delta V required to catch up with these things would also be an appreciable percentage of the speed of light. The star’s gravity would help, but it’s only going to take the edge off what will be an immense acceleration budget. Given that we basically haven’t hit even 0.01C, getting a planetary mass up to this speed seems impossible.
Plus, once you get wherever you’re going, you’d have to pay back whatever boost the star’s gravity gave you as you both escape its gravity well and spend down the velocity that the star itself was going.
Fuel, no. You’d spend the exact same amount of fuel catching up to this star and eventually leaving its orbit as you’d spend just travelling across the universe alone at the same speed. You’d only save fuel if you used the star for a slight shot maneuver, where you steal a tiny amount of its energy to accelerate you, but this would preclude you spending the voyage in the orbit of this rogue star.
If your fuel is something that is inside the star (or star's orbit), and you have a means of extracting it, then you can use the star as a refueling point.
You're right that using the star wouldn't require less fuel overall, but it could provide you access to extra fuel along the way.