Am I right to assume that the (implied) subject in the first part (up to 感じられ) of this sentence is the speaker and he "does the feeling"? If so, why would it be passive?

Not quite. If you take a look at the first clause

たった１分がものすごく長く感じられ

たった一分 is the subject here. The passive form is used here to say that the minute was felt (by the speaker) as a long period of time. It's similar to saying ピザが食べられた in that the person doing the acting is only implied. As for why the author chose to use the passive form here, to me it sounds slightly more natural than saying something like そのたった一分をものすごく長く感じた and like there's more of an emphasis being placed on "たった一分." Additionally, it sets up for the next clause where time is still the subject and is clearly acting as an agent against the speaker.

Am I right to assume that the (implied) subject in the first part (up to 感じられ) of this sentence is the speaker and he "does the feeling"? If so, why would it be passive?

Not quite. If you take a look at the first clause

たった１分がものすごく長く感じられ

たった一分 is the subject here. The passive form is used here to say that the minute was felt (by the speaker) as a long period of time. It's similar to saying ピザが食べられた in that the person doing the acting is only implied. As for why the author chose to use the passive form here, to me it sounds slightly more natural than saying something like そのたった一分をものすごく長く感じた and as if there's more of an emphasis being placed on "たった一分." Additionally, it sets up for the next clause where time is still the subject and is clearly acting as an agent against the speaker.