Learn English – The pronunciation of the word “honest” is with “h” or without

The critical question here is How does your teacher pronounce intervocalic /r/?

In some quite prestigious dialects—including the Stage British I was taught—/r/ falling between two vowels is pronounced as a voiced tip-flap. That is, the tongue does not curve backward toward the roof of the mouth but touches the back of the upper teeth, or the alveolar ridge immediately above that point, once, very briefly.

The sound which this produces is virtually indistinguishable from the alveolar flap which most Americans use for intervocalic /t/, as in beautiful. Indeed, I am not at all confident that these are two different sounds. The alveolar flap is notated in IPA with the 'fish-hook r', /ɾ/.

Wikipedia notes—I have no idea whether this is directly relevant to your situation—that

This sound is often analyzed (and therefore transcribed) by native English speakers as an 'R-sound' in many foreign languages. For example, the 'Japanese R' in hara, akira, tora, etc. is actually an alveolar tap^∗. In languages where this segment is present but not phonemic, it is often an allophone of either an alveolar stop ([t] or [d]) or a rhotic consonant like the alveolar trill or alveolar approximant.

^∗Note Wikipedia's assertion that “The terms tap and flap may be used interchangeably”.

The two are fairly interchangeable, in my opinion as a native [British] speaker. When emphasizing, we tend to use use [eɪ] more often than when not emphasizing. [ə] is more common in general, but really there's not much in it. [ə] can be dragged out more than [eɪ] when hesitating.