While doing transfinite induction to prove that open long ray is path connected, I got this conclusion that for any ordinal α, αX[0,1) (in lexicographic order topology) is path connected.
It seemed a bit against my intuition on path connectedness and I wonder if I have done something wrong or the statement is indeed true
No, this is not true when $\alpha>\omega_1$.
(Consider a path $\gamma$ from $(0,0)$ to $(\omega_1,0)$, and let $t_0$ be the infimum of parameters $t$ such that $\gamma(t)=(\omega_1,0)$. In the interval $[0,t_0)$, the path must fall within the long ray, but then the sequence $\gamma(\frac{n-1}{n}t_0)$ cannot converge to $(\omega_1,0)$, contradicting that $\gamma$ must be continuous).