Single pair shortest path algorithm with time a constraint
Your problem is NP-hard: The partition problem reduces to it as follows:

Let $ S = \{s_0, ... , s_{n-1} \} $ be a multiset of non-negative integers.

Construct cities $ \{ c_0, ... , c_n \} $, with each city $ c_i $ having 2 roads to $ c_{i+1} $, one with duration $ 0 $ and cost $ s_i $, the other with duration $ s_i $ and cost $ 0 $.

Then find the minimum-distance path from $ c_0 $ to $ c_n $ with total duration at most $ \frac 1 2 \sum S $.  This path has equal cost and duration iff $ S $ can be partitioned into subsets of equal sum.