Learn English – “run the opposite direction” or “run in the opposite direction”

I think the preposition you want to use is on.

To be at or by someone's side usually means a close, physical presence. It conjures up images like this one, where three friends are walking side-by-side:

To be on someone's side, though, usually means to be siding with them in an argument or dispute, or maybe supporting them in a tough situation.

I have heard both and I don't think either is incorrect. They both indicate an estimate of time, but to me one sounds loose than the other. When speaking about future events you may have an approximate time in mind or no set time at all, but when talking about a past event there is always a specific time even though you may not know what time that was.

An example of a future event:

-When do you think you will arrive?
-I hope to arrive around 7pm.

This sounds very loose to me. 7:30 is "around 7pm" but is equally close to 8pm. Hearing this would make me think the speaker has not thought too hard about time that they might arrive.

I should arrive at around 7pm.

This sounds more specific to me. Saying you will arrive "at around" a particular time sounds to me like the speaker has a specific estimate in mind but is allowing a little slippage either way.

When referring to a past event though, it seems more common to use "at" because this acknowledges that something happened at a particular time, you just do not know exactly when. This is commonly used in news reports, for example:

The accident occurred at around 3.30 p.m.

I don't think there is a right or wrong, both are probably idiomatic and used to mean the same thing, but certainly, with past events I would take note of the way it used in reporting news as they tend to observe consistent rules of grammar.