No, not in a mere few centuries. We need to become an interplanetary civilization first, able to travel within our own solar system with relative ease. That transition may happen within the next few centuries, for sure. Then our next big step will be to become an interstellar civilization, able to hop from star system to star system, which may take 100's of thousands of years, if not millions. It's only at that point we may have a hope of saving the Sun.
One possibility of prolonging the life of the Sun is to split the Sun up into a bunch of smaller stars. Smaller stars live longer. A red dwarf star only a 10% of the mass of the Sun can live 3 trillion years! Even if we say split the Sun up into 2 stars, 50% of the mass of the current Sun, then that would extend it's life to 56 billion years, instead of just 10 billion years as it is now!
How would you split the Sun up? A possible technology of the distant future would be to build a magnetic ring that orbits closely above the Sun and uses magnetic fields to siphon off the material of the Sun, as it boils and bubbles away from the Sun. The siphoned off material would then get deposited on the other side of the magnetic ring, and it would start creating a new protostar. Initially the protostar would look like a bulge on the side of the Sun as material gets continually transferred from Sun to protostar. Of course, by that point, the Sun would be producing far less heat, and so would these new protostars, so you'd have to move the Earth closer to these new stars to get the same amount of heat. Should not be a problem for a civilization able to create new stars from parts of an existing star!
You might even consider creating new Earths while you're at it, to increase living space. First you might want to move Venus into one of these stars' habitable zones, and then colonize Venus, which would be very Earth like by then. Perhaps move Mercury into orbit around Venus to stabilize its orbit, like the Moon does for Earth.