So, using independence of the increments, I'm tracking $P[X_{3} < 2 | X_{1} = 1] = P[X_{3} - X_{1} + X_{1} < 2 | X_{1} = 1] = P[X_{3} - X_{1} + 1 < 2 | X_{1} = 1] = P[X_{3} - X_{1} < 1]$ and this last expression is equal, in distribution to $P[X_{2} < 1] = P[X_{2} / 2 < 1/2] = \Phi(1/2) = .69$
Am I missing something? I'm pretty confident this is correct.
Almost -- from independent increments
$$\forall t>s\;\;(X_t - X_s) \sim N(0,t-s)$$
For our problem: $$t=3, s=1 \implies X_3 - X_1 \sim N(0,2) \implies X_3|X_1 \sim N(1,2) \implies P(X_3<2|X_1=1)$$ $$ = \Phi \left(\frac{1}{\sqrt{2}}\right) \approx 0.76$$