Consider a Poisson process for which events occur at a rate of $3$ per hour, and let $T$ be the time in hours to the $2^{nd}$ event. What is the value of $P(T≤0.5)$?
I think this would be the same as the probability of having at least $2$ occurrences in a half hour with mean $\lambda$ $=$ $3(.5)=1.5$
We have, $$P(X\geq2) = 1 - P(X=0) - P(X=1)$$
$$= 1-\frac{1.5^0e^{-1.5}}{0!}-\frac{1.5^1e^{-1.5}}{1!}$$
$$=.442$$
Is this a valid solution or do I need to do some kind of integration?
Yes. your working is fine. No integration is needed.
Note that the time until the second event is also known as the Erlang distribution and you have just rederived the idea to get its CDF.