Yes, 'so much' can certainly be the subject of a sentence, and the sentence you've quoted is a common and natural example.
Your understanding of it is correct: it means 'so many things have happened to us so fast'.
In this context, 'so much' is better than 'so many things'. If you say 'things', it makes us think of a number of separate events. 'So much' is broader: it suggests the cumulative effect of these events, including less tangible experiences, such as emotions you have felt and changes that have occurred.