Hi Sigurd and welcome to the forum,
For someone short on experience, you certainly aren't short on common sense because you've just perfectly highlighted one of what I would consider being a negative in regards to dealing with a drop-shipping model.
There is quite simply one way and one way only that I'm aware of that you can do to try and mitigate any customer service issues that you may encounter and that is to have seamless access to your supplier.
By seamless access I mean have a supplier that will provide you with either an account's manager to service you or a direct line you can call to get a speedy resolution to your problem. Plenty of suppliers will prefer to have you email, I would suggest that if your concern is customer service, you should avoid suppliers that will only offer that as support.
You need open and fast communication channels to get fast resolutions to problems outside of your control. So when speaking with suppliers, put that high on your list. If they can't provide that level of customer service to you, just imagine what level of service you will be able to provide your customer ..........