There are two major issues here - shock and forward planning.
“Close” relatives are not necessarily emotionally close; indeed it is sometimes almost impossible to maintain emotional bonds over long distances for sustained periods. Yet it is also true that news of the imminent or recent death of a “close” distant relative can come as a real blow. It is as though the extinction of a familial tie sharpens its value.
That suggests that it is a mistake to think that “they haven’t seen each other in years so he’s not going to be much bothered.” It is safer and kinder to assume that “he” will be considerably “bothered” and therefore needs the news breaking gently and supportively. That usually implies a telephone call, because that allows you to lead cautiously to the hard news and to respond to the recipient of your call as the news sinks in. No one likes making such calls, and it is all too easy to persuade yourself that a text or email or even a letter would be acceptable. Acceptable, perhaps; but not compassionate.
Lead to the hard news slowly. “He” may not often hear from you and he will be wondering why you are ringing. To that extent he is preparing for bad news. Give him a little time. He may well ask after the deceased quite early in the conversation. “And how’s Mary?” Again play for (a little) time. “Well, I thought you would ask that. And of course it’s about her that I am calling. You know she has not been well for some weeks now. She was taken into hospital ten days ago and gradually went downhill……” Now you have prepared him. You have given him time to begin to adjust. He’s already guessing what you are going to say. So say it.
People vary hugely in the way they react to unexpected bad news. Don’t take any reaction at face value. And certainly don’t take offence if it is hard to hear. Don’t prolong the conversation, but make a point of ringing back the next day to check on how “he” is coping with the news.
That raises the second issue: forward planning. “He” may well feel that he should attend the funeral, and so will want to know when and where it is to be held. Sometimes, it takes several days to establish that, especially if it is a cremation at a time of year when crematoria are unusually busy. “He” may find the uncertainty hard to bear and he may get grumpy with you because you cannot yet give him a definite date. Bear that as well as you can. If he wants to send flowers to the closest relation(s) or have a bouquet on the coffin, the undertaker can organise both of those requests. Indeed if “he” seems to want to be closely involved, put him in touch with the undertaker. His staff will be well used to liaising with family members: it is likely you will have plenty of other things to be doing.