I can find no RAW, or any rulings at all, that can help answer this question. That being said, the following is speculation based on the entry on PHB page 205 that details ‘The Weave’ of magic.
In conjunction with how Wizards learn their magic, and the knowledge of the Arcane arts and the understanding they must have of the Weave in order to cast such magic, we can assume that yes: A wizard would know when the magic he cast a few turns ago that affected the Weave in such a way as to turn an ally invisible has lost its effect and thus become inert. I think this particularly holds true for concentration spells where the Wizard is actively maintaining the alteration to the Weave in order to sustain an effect.
When the spell ends, either through the Wizard deciding to end it, the spell running its full duration, or something ending it prematurely, I have no reason to believe this would be lost on the Wizard in question.
The bless spell is not interrupted
The key to the answer is:
You briefly stop the flow of time for everyone but yourself.
For everyone else, delta t is zero.
No time passes for other creatures
Turns / rounds each last six seconds -- six seconds elapse from the beginning of someone's turn and the beginning of their next turn. From this game mechanic, we can say that the usual delta t from round to round is six seconds. Spells with duration of a minute can be modeled as being on a counter in this case: you count down one for each turn/round elapsed.
For everyone who is not the archmage, their time / turn / round counter remains where it was when time stop went into effect. Since no time is passing for them, the durations of their spells are, compared to the archmage's frame of reference, extended. The archmage's delta t keeps moving. For the other characters, they don't notice a difference since for them time stopped.
The bless spell's duration depends on the caster
In the case of bless, a concentration spell, the duration is keyed to the character who cast it, which is also the character who is concentrating on it. If it is a character who is not the archmage, their delta t is zero so their duration counter resumes on their next turn. (That is after time stop ends). Their spell still lasts a minute for them because time (and the counter) stopped for everyone but the archmage. That would provide the archmage with a few extra rounds / turns of the bless spell's magical effects.
If the archmage cast bless (from a ring of spell storing) it's duration would keep decreasing since time/turns are ticking away for the archmage.
@Medix2 asked for clarification in a comment:
If bless were cast by somebody else, but cast on the Archmage, would they benefit from it?
The answer is: yes. Nothing in the time stop spell says that it stops the effects of other spells. Bless, once it takes effect, keeps on taking effect until the spell ends (or concentration ends). Nothing in its text says otherwise.
Concentration
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends. (Basic Rules p. 79)
In plain English, you in those two sentences is the caster of a spell requiring concentration. Unless you lose concentration, the spell does not end.
Best Answer
As I was still searching for the answer I found this:
Timed Durations
In the SRD