I did it on dynamic IP connections with far less upstream for more than a decade. Luckily I've had 5-10 megabit up for another decade on top of that and it's even easier now. But it's certainly possible and even easy on limited connections. We're not talking 56k here. Also, limiting bandwidth to some appropriate value is also not hard.

And so what if your websites goes offline for a few hours a handful of times a year? Or even a week. It doesn't matter. This isn't some profit driven job.