Learn English – word for immediate future just like “just” for immediate past

Yes, they are both considered conjunctions. Furthermore, the two expressions are synonmyms.

I think the reason for you having difficulties with these sentences is that some information has been ellipted:

I am no one's spokesman, let alone his.

is the same as:

I am no one's spokesman, let alone that I would be his spokesman

And:

I hardly have time to think these days, much less relax.

is the same as:

I hardly have time to think these days, much less do I have time to relax these days.

The context tells you whether or not N hours ago is precise. I would say that, typically, it is not. To be clear, you could add about to the expression:

about N hours ago.

Otherwise, the context will help you determine that it is precise, or you might add exactly to the expression:

exactly N hours ago.

I don't think your usage of N hours ago is correct. For example, if it's 8:40, and you say something with one hour ago, then I don't think it is perceived as anything between 7:00 and 7:59. Instead, it is understood as 7:40 plus or minus a few minutes. In other words, the approximate interval is centered around 7:40 and it is not the interval 7:00 to 7:59.

As for your particular examples, I don't think there's a rule. You are just going to have to choose what feels best.

If you really want to use some rule, you could round to the closest hour. So if something happened 1 hour and 20 minutes ago, you would round down to 1 hour and say X happened 1 hour ago. So for your first example, you could say 1 hour ago, and for the second, you could say 2 hours ago.

Including something like about, close to, almost, etc also sounds natural. Using your numbered examples:

1. I got here about an hour ago.
2. I bought this almost two hours ago.