Is there any way to get continuation for /api/0/stream/items/ids like /api/0/stream/contents/?
If no, how can I get next item-ids of a stream? Or is there any way to convert item-id/timestamp to continuation?
The /api/0/stream/items/ids endpoint supports an nt parameter that you can use to get increasingly older items. You can take the timestamp (in microseconds) of the oldest item that you already have, convert it to seconds, and then pass it in as a the ot parameter. See this page for the full list of parameters that are supported.