As I understand it, you are asked for FTP details in case of a permissions issue. It may or may not have anything to do with your host.
There is nothing wrong with you trying to use the auto update again. If that doesn’t work then give the FTP info requested.
If that becomes a pain for you, Google “update asks for ftp”. You will get a number of options to solve the problem. But all will require you to add something using FTP to various files. Usually wp-config.php.