Que is a causative conjunction here. For the use of que instead of y, I'll be referring to §2.3 and 2.4 of the entry about que in the DPD (Diccionario Panhispánico de Dudas). Translating and summarizing, these say that
- Que introduces consecutive subordinate propositions, commonly coordinated with the adverbs tan(to) or tal (English: "so many, so much, such");
- In some colloquial sentences the first part (with tan(to) or tal) is omitted because it's more or less obvious.
The "complete" sentence in Spanish should then read
Me dio tal aventón que por poco me caigo.
or a variant thereof like
Me dio un aventón tan fuerte que por poco me caigo.
This is exactly parallel to the English usage or such/so many/so much... that..., which would make the correct translation the following:
He gave me such a push that I almost fell.
So que in the original sentence serves the purpose of suggesting the cause of the fall. It's not that you were pushed and then you fell; it's that you were pushed so hard that you fell because of it.
As for the present tense in ...por poco me caigo, my answer to this question deals with it (the expression is not the same but the grammar is). Summarizing, it seems to be a phenomenon called presente de conato. It's the use of the present tense to express an action that would have taken place in the past, and maybe started or seemed to start to develop, but didn't actually come to pass. It nearly always comes together with adverbs and phrases like casi "almost, barely" or por poco "by a small amount".