If we accelerate a neutron star to move at speeds comparable to light would it become a blackhole because of the added energy? Could a neutron star turn into a blackhole if it is accelerated to near luminal velocities? And if it slows down after a while, does it become a neutron star again?
In special relativity, changes of reference frame are symmetric between observer and observee.  If moving a neutron star gave it enough kinetic energy to become a black hole, then you could effect this change by jumping into a moving reference frame yourself.

What matters is the neutron star's mass in its rest frame.