The product owner role has a large number of responsibilities, which can be broken down into tasks that can actually keep the product owner-occupied for full-time. Ideally, it is not ADVISABLE to have a product owner is not available for the team HALF of the time. But in CASES where they have a Part-Time product owner, they have to support a lot. There will be a few tasks which the team has to take over, like BRAINSTORMING alone as a team without the product owner and coming up with the queries. 

They might even have to do follow-ups which could have been taken care if they had a full-time product owner. Part-time product owner also becomes a challenge as the timely feedback from the client might not be possible all the time. In such cases, the team can take help from the BA (Business analyst) who can work as a proxy to help the team move forward. The BA can connect a regular interval to seek guidance from the Product owner. Usually, in cases where the TEAMS do not have their product owner co-located, they take support from the BAs.