Using Online Etymology Dictionary as a source, it seems that in most examples the initial 'a' in these words are from roughly the same morphemic meaning, with the exception of "amount". I've quoted the meaning I think is relevant to your words.
a- (1) In words derived from Old English, it commonly represents Old
English an "on, in, into" (see on (prep.)), as in alive, above,
asleep, aback, abroad, afoot, ashore, ahead, abed, aside, etc.,
forming adjectives and adverbs from nouns, with the notion "in, at;
engaged in." In this use it is identical to a (2).
"away or apart from a normal direction or position, out of
the way," from a- (1) "on" + side (n.).
from Old English an, on (see a- (1) + like (adj.)
c. 1200, "in life, living," contraction of Old English on
life "in living, not dead," from a- (1) + dative of lif "life" (see
"on from this (that) place;" see a- (1) + way (n.)
from phrase on round; see a- (1) + round (adj.).
The exception seems to be "amount"
a contraction of the prepositional phrase a mont, from a
(from Latin ad "to;" see ad-)
Where the a- prefix seems to have been originally from Latin ad- (to/towards) which went through Old French and became a-.
In Old French, reduced to a- in all cases
So it seems that in most of your particular cases the a- prefix seems to have come from the first meaning listed.
However take note that I only mentioned one meaning of the Old English a- prefix, because this was the one that seemed applicable to your words.
As the Online Etymology Dictionary says, in words like anew, afresh, akin and abreast the a- has probably a meaning of:
Middle English of (prep.) "off, from,", and in
And in words like abide, arise, awake, ashamed the a- is either:
Old English intensive a-
probably implying originally "motion away from"
marking a verb as momentary, a single event.
Given that the initial and pften distinguishable a- part of the word can, or at least used to mean so many different things, it's not surprising that The Etymology Dictionary includes this note from the OED:
[I]t naturally happened that all these a- prefixes were at length
confusedly lumped together in idea, and the resultant a- looked upon
as vaguely intensive, rhetorical, euphonic, or even archaic, and
wholly otiose. [OED]
As for the name of this a- prefix, I have no idea, I'm sorry. From what I read I only know it as the Old English a- prefix or particle. I doubt that a name for this would be particularly helpful from a descriptive point of view, because it just can mean so many things. Compare this with the Greek privative A below, which is very consistent in its meaning and does have a name.
As for what phenomenon this is, I can't think of anything more general than "prefixation" (adding of a prefix to a word).
One thing the a- prefix doesn't mean in your words is what the a- prefix means in "atypical", "apolitical", "atonal", "achromatic". The a- here is completely different from any of the words mentioned above. It's Greek meaning negation or absence.
Alpha privative or privative A
Also, to answer your question:
Is there a general rule to know the meaning of a prefixed word from
its base word without using the dictionary?
In the case of this Old English a- prefix, definitely not. However there are other prefixes which are much more consistent. If you see a word beginning with un- or in- followed by a word separately recognisable you can have an idea it means its opposite or its negation. There are exceptions though, like the famous flammable/inflammable, or even "famous" and "infamous"; "infamous" meaning you're still famous but for a different reason.
So you can pick up general rules, but I don't know of anything bulletproof.