Answer
The time spent in field $B$ would have been equal to $T_0$
Work Step by Step
We can write an expression for the radius of curvature of the particle:
$r = \frac{mv}{\vert q\vert~B}$
If the velocity is $0.5~v_0$, then the radius of curvature is half of the original radius of curvature.
Then the distance traveled inside the magnetic field would be half of the original distance.
Traveling half the original distance at half the original speed results in the same time $T_0$
The time spent in field $B$ would have been equal to $T_0$.
