The Gamma-Poisson cousin

You model an event rate $\lambda$ (events per hour) with a prior $\lambda \sim \text{Gamma}(\alpha, \beta)$, where $\alpha$ counts prior events and $\beta$ counts prior hours, so the prior mean is $\alpha/\beta$. Take $\alpha = 2$ and $\beta = 1$ (prior mean $2$ per hour). You then observe $18$ events over $6$ hours.

What is the posterior, its mean, and how does it compare to the raw rate?