We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory of $O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$, where $|M_G|$ denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Bernstein, 2015], which achieves a $3/2 + \varepsilon$ approximation for the maximum cardinality simple matching. When $W$ is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a $2 - δ$ approximation (for some small constant $δ\sim 10^{-17}$) for the maximum weight simple matching. In particular, for the weighted $b$-matching problem, ours is the first result beating the approximation ratio of $2$. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Bernstein and Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a $2 - \frac{1}{2W} + \varepsilon$ approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry's duality theorem.