The order of ε-transitions in NFA
I think the answer is somewhat subtle and lies in Sipser's definition of what it means for an NFA to accept a string. 


  Let $N = (Q,\Sigma,\delta,q_0,F)$ be an NFA, and $w$ a string over the alphabet $\Sigma$.
  Then we say that $N$ accepts $w$ if we can write $w$ as $w=y_1y_2\dots y_m$, where each $y_i$ is a member of $\Sigma_\varepsilon$ and ...


Compare this with his definition of what it means for a DFA to accept a string:


  Let $M = (Q,\Sigma,\delta,q_0,F)$ be a DFA and let $w=w_1w_2\dots w_n$ be a string where each $w_i$ is a member of the alphabet $\Sigma$. Then ...


Therefore, by his definitions, his NFA $N_4$ accepts $a$ as it can be written as $a=\varepsilon a$, and $\varepsilon$ and $a$ are both members of $\Sigma_\varepsilon$.