Your questions could be valid for any time machine regardless of its base, you do not have to restrict yourself to cosmic strings only.
However, the problem is that travelling in time (at least backward) is forbidden by causality principle. Tranveling to past could lead to many paradoxes, for example grand father paradox (what would happen if you kill your own grandfahter - you have never been born, hence you could not have killed the grandfather).
The problem is also second law of thermodynamics. Going backward in time requires to decrease an entropy in whole universe. To do so you need to do a work and increase entropy somewhere else, however where (in some other space for example but theories of multiverse are rather speculative and often unfalsifiable)?
In principle, it is possible to travel to future. If you move with a speed very close to that of light, thanks to time dilation you will arrive back to starting point in the future. But you cannot return back in time for reasons I described above.
There are other theories concerning for example worm holes allowing you to traverse space and time instantly. However, they are connected with negative energy densities which are forbidden by general realtivity (see energy conditions).
Concerning the Alcubierre drive. It also operates with negative energy density. There are several other proposal how to reduce energy high negative energy requirements and even metrics where only positive energy are needed. But all these constructions operate with already existing warp bubble. So far, no body has been able to propose how to construct the bubble and make it moving (see here).