Random intercepts with high-cardinal categorical feature?

The use case is optimizing decisions for each device. For example, if we want to create some kind of send time optimization process to e.g. deliver push notifications at the time a user is mostly to be engaging with their device, we would want to estimate the most likely time of engagement for a particular user. For users with a lot of engagement history, we would want that estimate to lean more on their own engagement. For users with less (or no) engagement history, we would want to shrink the estimate towards the average. This way we can get progressively more personalized estimates as users engage with their device more while maintaining our “best guess” for those that have little or data.