One should not expect the reaction of the reader to be the same as that of Emily Dickinson's speaker since the reader is aware that it is Death who arrives for the speaker in the first line while the speaker is not.
On the contrary, the speaker perceives Death simply as a gentlemanly carriage driver as evinced when she is completely taken by surprise upon realizing that the horses are headed "toward eternity." The poem's ending, therefore, is an example of situational irony, an irony that occurs when there is a discrepancy between what is expected and what actually happens.
For, Dickinson's speaker is flattered when the carriage driver arrives and wishes to take her on an unexpected ride through life past schools and fields of grain, a ride she feels is merely a leisurely one.
We slowly drove--He knew no haste
And I had put away
My labor and my leisure too,
For His Civility--
However, this long ride turns out to be a trip of eternal significance as the speaker "first surmises" that Death is present as she has passed a "House" whose cornice is in the ground (a grave) and the driver has turned the horses toward "Eternity."