Weak one way functions from strong one way functions
No, $g$ is certainly not even weakly one-way. In particular, in order to invert $g$ you just need to be able to sample a $y$ such that the first bit of $f(y)$ equals the first bit of $f(x)$. Now, since one-way functions work on random input, it is clearly not hard to sample an input with the same first input bit as $f(x)$, since $x$ was randomly chosen.