The long line is $S_\Omega \times [0,1)$ (not $(0,1]$) in the dictionary order topology (see p. 158 bottom (ex. 12) (all references to the 2nd edition of Munkres))
Then ex. 6 (chapter 3) on that same p. 158 says that if $X$ is well-ordered then $X \times [0,1)$ (in the dictionary order) is a linear continuum. So the long line is a linear continuum.
Finally ex. 8 on p. 206 (chapter 4) says that a linear continuum is normal.
So the long line is normal.
(Side note: it's not too hard but non-trivial, hence Munkres leaves it out of his text book for that reason I think, that any linearly ordered set in the order topology is even monotonically normal, which implies it is hereditarily normal (or completely normal), and this even extends to so-called generalised ordered spaces (which includes the lower limit topology as well). Now the result seems to depend on well-orderedness etc. while in fact it does not, really).
The long line is not metrisable, because it is limit point compact but not compact (for metrisable spaces these notions are equivalent). That's one of the possible arguments for it.