Two things:
Yes, $x_1$ is the first bit. The idea is that if $x_1 = 0$ (which occurs with probability 1/2), it is simple to find a preimage of $g(x) = 0$ --- any string $x'$ with $x'_1 = 0$ will suffice. This shows that $g$ cannot be an $\alpha$-OWF for any $\alpha <1/2$. To show that it is an $\alpha$-OWF for $\alpha \leq 2/3$, you would need to reduce to the strong OWF security of $f$, which I will leave for you to do.
The choice of $2/3$ is simply a social convention for a "suitable constant". There are many complexity classes $\mathcal{C}$ that depend on some parameter $\alpha$ (which I will denote $\mathcal{C}(\alpha)$) where you can show some result of the form
For any $\alpha$ bounded away* from $1/2$ and $1$, the complexity classes $\mathcal{C}(\alpha)$ are equivalent.
Here, "bounded away" means that $\frac{1}{2}+\frac{1}{n^c} \leq \alpha \leq 1 - \frac{1}{n^d}$ for constants $c, d$ --- in particular, $\alpha$ cannot be negligibly close (as a function of the size of the input) to either 1/2 or 1. For such classes, the social convention to choose $\mathcal{C}(2/3)$ as the "standard" example to relate things to is common.
There are many examples of the above phenonoma, for example most randomized complexity classes, but perhaps $BPP$ in particular is the best-known example.
The importance of $\alpha$ being bounded away from 1/2 and 1 can be seen via the difference between the classes $BPP$ (which has this restriction), and the class $PP$ (which doesn't, and is much more powerful).
Anyway, this section of the linked notes are essentially showing that one-way functions are a similar class to things like $BPP$ (in terms of their dependence on a parameter $\alpha$).
