It's not necessary to determine the absolute configuration of the molecule before S_N2 attack, but I've included it for clarity's sake.

The small numbers one through four refer to the CIP** priorities of the branches.

After S_N2 substitution, the molecule has the following conformation:

Attack has occurred with inversion of configuration, and the absolute configuration has changed from "R" to "S."

The nucleophile, tert-butoxide, must get to the σ C-Br antibond for an S_N2 reaction to occur. The path to the antibond is blocked by the bulky tert-butyl groups attached to the stereocenter. The nucleophile also has a bulky ter-butyl tail. It cannot reach the antibond due to steric clash with the tert-butyl groups. This effect is explained in the $\mbox{S}_{\mbox{N}}2 \mbox{ vs. E}_2$ section of this SparkNote.

A will undergo a faster S_N2 reaction. A has a more stable S_N2 transition state because the partial negative charge on the α -carbon will be hyperconjugated** into the ring's Π electron system.

More stable transition states translate into faster reactions. Thus A is faster than B.