I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review [here][1]). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in [Zygmund: Trigonometric Series I][2].

**Added.** I am not so sure that a simple solution for part (b) exists, given that it implies Theorem B in [Zygmund's original paper][3], which is already rather deep. (Actually, Zygmund remarks in the beginning of Section 2 that for the existence of radial limits the proof is essentially the same.) This paper is also the original source for the result I quoted from his book, see Theorem D there.

  [1]: https://zbmath.org/?q=an:54.0232.03
  [2]: http://www.isbnsearch.org/isbn/0521890535
  [3]: http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1619.pdf