The min-diameter of a directed graph $G$ is a measure of the largest distance between nodes. It is equal to the maximum min-distance $d_{min}(u,v)$ across all pairs $u,v \in V(G)$, where $d_{min}(u,v) = \min(d(u,v), d(v,u))$. Our work provides a $O(m^{1.426}n^{0.288})$-time $3/2$-approximation algorithm for min-diameter in DAGs, and a faster $O(m^{0.713}n)$-time almost-$3/2$-approximation variant. (An almost-$α$-approximation algorithm determines the min-diameter to within a multiplicative factor of $α$ plus constant additive error.) By a conditional lower bound result of [Abboud et al, SODA 2016], a better than $3/2$-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time. We also present the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph.