According to CGEL, dropping the article of count nouns in the singular can happen (outside 'headlinese') under only restricted circumstances. For completeness, I will reproduce the full discussion. Your first example belongs to the category [20vii] ('matched nouns'), while your second example belongs to [20vi] ('repeated nouns').
One caveat is that in your second example, what is repeated is a two-word noun phrase (NP) rather than a single noun. But what rules of syntax concern almost always are not individual words of a given lexical category (e.g. 'nouns'), but rather structures that have the same syntactical function (NPs, regardless of whether they consist of a single noun or a noun plus dependents).
In CGEL's terminology, if a 'noun group' is not preceded by a determiner, it is called a 'bare NP'. Other sources might say that it has a 'zero article'.
Note that the asterisk (' * ') in front of a group of words indicates that this group is not acceptable as a sentence or part of a sentence.
So here is CGEL, on the subject of bare NPs (pp. 409-410).
8.5 Restricted non-referential interpretations of bare NPs
Here we cover cases similar to those dealt with in §8.4, i.e. cases
where only a restricted range of head nouns is found. This time,
however, our concern is with bare NPs. We confine our attention to
singular count nouns, which normally require a determiner.
(a) Bare role NPs
 i Henry became treasurer.
treasurer, I strongly support this proposal.
iii The role of
treasurer will fall to Henry.
NPs such as treasurer, deputy leader of the party, occur as the
predicative complements of verbs like be, become, appoint, elect, as
oblique predicative governed by as, and as complement of the
preposition of following nouns like role, part, or position. They
cannot occur as subjects (*I'm told treasurer
strongly supports this proposal) or objects (*We
dismissed treasurer): in these positions they require a
determiner. The interpretation of bare role NPs is invariably
definite: in , for example, we are concerned with the office of
treasurer in some particular organisation. They are therefore
invariably replaceable by their counterparts with determiner the, as
in Henry became the treasurer, and so on.
It is important to
note that only NPs which genuinely allow role interpretations can
function in this manner. The verb become, for example, also allows
non-role nouns such as miser as head of the predicative complement,
but such nouns require determiners: Fred became
*miser/a miser when he lost his job.
(b) Fixed expressions or frames
 i Ed is in hospital / went to
school / went off stage. [activities linked to
are out of place / off target / on
went by bicycle / communicate by
iv We had
lunch on the terrace. [meals]
dawn, by daybreak, before sunrise [times]
in arm, back to back, day after day,
mouthful, side by side, mile upon mile nouns]
father to son, from beginning to end, between [matched
and wife, mother and child nouns]
The examples in [i] are illustrative of a number of restricted
expressions connected with common activities of everyday life where no
determiner is permitted, even when the noun involved is in other uses
a singular count noun denoting a location which would require a
determiner. In these cases, the noun acts as an indication of the
associated activity, and does not have its standard denotation.
Contrast, for example, Ed is in bed (resting/asleep) with
referential There are fleas in the bed; or Ed is in prison
(serving time) with There was a riot in the prison; or again Ed is
in church (at a service) with There is a new pulpit in the church.
The nouns which permit this use are severely restricted. We do not
have for example *Ed is at desk (studying),
*Ed is at computer (working), *Ed is
in kitchen (cooking). Nevertheless there are a fair number of them:
others include the underlined [here boldfaced] nouns in such
expressions as on campus,in class, at college, settle out
of court, at sea (as a sailor), at table (for a meal),
leave town, start university.
The examples in [20ii] relate to what we may call 'status': whether or
not something is in its proper place, whether or not someone is
available or engaged in their proper activity. These fixed expressions
are comparable to those with non-count nouns such as at work, at
play. The bare NPs in [iii] occur after the preposition by: compare
the definite NPs in [i7iv]. Meals are generally expressed by bare NPs,
except when a particular occasion is singled out: compare [20iv] with
We had a nice lunch at the Savoy. Note also at/by dinner, where the meal indicates a time of day. Bare NPs are used for times of
day following the prepositions at, by, before, until, after, as in
[20v]; note, however, that morning, daytime, evening, and dark
take in + the instead of at: in the / *at
morning. The examples in [vi-vii] are illustrative of anumber of
expressions involving repetition of the same noun or contrasting
nouns; for thecategory status of the expressions in [vi], see Ch. 7,
§4.3. Similarly, in coordinate structures, bare NPs can optionally be
used in repetition: We searched endlessly for a spring or a
cave to spend the night, but neither spring nor cave could
On the subject of bare coordination, see also this answer.