The sentence endings in Korean may be classified as follows.
1. The common standard endings with no special connotations (the most important ones).
- ~ㄴ/는다(v), ~다(a) (먹는다, 예쁘다) - formal, plain. (To someone senior, in formal setting)
- ~ㅂ니다(v, a) (먹습니다, 예쁩니다) - formal, polite. (Impersonal, as in novels, newspaper articles, etc)
- ~아/어(v, a) (먹어, 예뻐) - informal, plain. (Familiar speech between friends)
- ~아/어요(v, a) (먹어요, 예뻐요) - informal, polite. (Common form for most occasions other than between friends)
* plus interrogative endings ~가?, ~ㅂ니까?, ~냐?, ~니?, and ~아/어(요)?,
(~야/여 is a special case of ~아/어 for the special adjectives 이다(is) and 아니다(isn't))
2. Standard endings with special connotations: ~네, ~군, ~지, and many others.
3. Non-standard endings (the sentence ends in the middle with the trailing part omitted)
- ~ㄴ데 - when/as. "그런 뜻이 아니었는데 (오해 하시네)".
- ~지만 - even though. "물론 그렇긴 하지만 (꼭 그런 건 아니야)".
- ~아/어도 - even so/then. "아무리 그래도 (어떻게 그럴 수가 있어?").
(Almost all verb connective forms can be used this way, although it may not always be a good practice)
4. (Almost) Randomly rearranged phrases with no patterns - common in song lyrics and internet writings.
Because comments left at social sites are often as informal as day-to-day conversations, a lot of exchanges verge on casual speech giving rise to all kinds of ungrammatical phrases.
Your example "그때 기억해 초등학교 무렵" falls in the #4 category. As written, it is more like an informal utterance than a proper sentence. It can be written as "그때 기억해, 초등학교 무렵" (I remember those days, around the time of grade school), to make it clearer, but not everyone bothers to make the effort.
Such haphazard endings can be very confusing to learners. I think it's a waste of time to try to understand them.
It might even impede one's learning if the learner constantly exposes themselves to them.
The best way is to concentrate on textbook-like sentences until one has a firm grounding on sentential structures.