Words don't in general belong to a single part of speech (POS), that is POS is not in general a lexical property of a word, but they are normally associated with a particular dominant role/POS which in this (*day) case is Noun (or NP), although like any (prepositional) noun phrase, particularly one locating in space or time it can act as an adverb. Nouns, parts of the body, time points or durations, etc. can even act as verbs (when there is no more obvious and appropriate verb to use - he shouldered him aside and headed the ball into the goal; he minuted the discussion). The true English adverbs are a closed class or marked with a functional morpheme (usually -ly) - he sidled slowly, crablike, into the room.
Thus here... Yesterday, today and tomorrow are nouns that can act as complete noun phrases as they are reductions of forms that include a determiner. See e.g.
In general, (prepositional) noun phrases can act adjectivally and adverbially, and in particular when the noun is already marked for place or time, the preposition and/or article can often be omitted as definiteness is understood, particularly in sequences of places and/or events. In journalism and diaries, place, time and byline conventionally omit the propositions. It is not so much that there is anything special about *day (except you normally won't add another determiner to today/tomorrow/yesterday as complete deictic noun phrases centred on the time of speaking/writing). When you do add a determiner it recentres the deixis (which is an attribute describing a word that locates something in the space-time context).
- Sunday, I visited some museums.
- London, I was exhausted after the flight, but Edinburgh, I visited every museum.
London Sunday, another explosion rocked the underground as evening commuters...
all my tomorrows will be spent with you!
- every tomorrow is a new day!
- the tomorrow I'm looking for ...
my tomorrow isn't looking too promising!
all my yesterdays are as nothing now I have met you!
- yesterday evening's train ...
- last night's train ...
- yesterday's events ...
- the week's events ...
- this week's evening events ...
In speech when a multiword noun phrase is used as an adjective rather than a single adjective, the extra words are conventionally hyphenated prefixes in written form to show that the additional words are not modifiers of the main noun, but of the adjectival noun (unless there is no difference in semantics):
- early Iron-Age artefacts
- an in-your-face kind of guy
- an on-the-ball comment
- a last minute decision
The last example is sufficiently frozen that it is still common without the hyphen although it does now occur with the hyphen. The more extreme version of this wordifying push is when the hyphen/space gets dropped completely as has happened with today, tomorrow and yesterday. 30 years ago it was most often 'to-day', and 50 years ago even 'to day' and 100 years or more ago 'the day' or 'this day' (sufficient unto the day is the evil thereof; give us this day our daily bread) - this has changed twice during my lifetime!