"그리워하는데도 한 번 만나고는 못 만나게 되기도 하고, 일생을 못 잊으면서 아니 만나고 살기도 한다. 아사코와 나는 세 번 만났다. 세 번째는 아니 만났어야 좋았을 것이다. 오는 주말에는 아사코가 살았던 춘천에 갔다 오려고 한다.소양강 가을 경치가 아름다울 것이다."
"You might end up never seeing the one you miss anymore after seeing them just once, or you might live a whole life never seeing the one you cannot forget. Asako and I have met three times. The third time, it would have been better if we hadn't met. This weekend, I am going to visit Chuncheon where Asako once lived. The scenery of the Soyang river there must be beautiful."
-고는 (만나고는) describes one event occurring before another one. It is similar to the conjunctive connector -고, except that -고 doesn't express the sequential aspect as clearly . For instance, 아이들이 큰 소리를 지르고 달려갔다 doesn't make it very clear whether they shouted first and then ran or did both at the same time. 아이들이 큰 소리를 지르고는 달려갔다 on the other hand makes it explicit that the shouting happened first. -고는 is thus like in between -고 and -고 나서 which makes the two events even more clearly separated in time.
아니 is the same as the verb and adjective negator 안. 안 is a contraction of 아니, so 안 가다 and 아니 가다 are exactly the same in meaning. 아니 is not used much nowadays but you still see them occasionally in writings.
EDIT (response to a comment):
일생을 in 일생을 못 잊으면서 아니 만나고 살기도 한다 can be confusing to learners because 일생을 appearing before 못 잊으면서 might look like they are talking about not forgetting their life (일생) itself, but that is not the case. The real object of 못 잊으면서 is not given in the sentence, but since the essay is talking about connections between people (인연), it is implicit in the overall topic and the sentences preceding this one.
I am not sure how one might classify 일생을 grammatically, but semantically it is like an adverbial phrase. It is similar to 일생 동안(에), but when you want to stress the degree, purpose, or some other aspect of it, you can say 을/를 instead of 에, 에서, and others. This happens a lot in Korean, as you can see in sentences like 길을 걸었다 (more natural than 길에서 걸었다) and 두 시간을 기다렸다 (not 두 시간 동안에 기다렸다). If you apply strict logic, 두 시간을 기다렸다 might be wrong because what you actually waited for is not the two hours but some person or thing, but in practice it is actually better in that it sounds more compelling while at the same time giving the sentence
better structural balance.
So 일생을 is more like an adverb that further details the rest of the sentence (hard to say exactly which of the three verbs it applies to, but I would say to all of them). To back it up, the sentence will still work (although not as good as the original) if you move it in front of one of the other verbs: 못 잊으면서 일생을 아니 만나고 살기도 한다, or 못 잊으면서 아니 만나고 일생을 살기도 한다.
To understand this type of sentences, it helps to consider the whole sentence when parsing it, keeping in mind the overall context.