Is it possible to host a dynamic map service in ArcGIS Online?

No. It is not possible to host a dynamic map service in ArcGIS Online. A map service is limited to ArcGIS Server. A map service must be published on an ArcGIS Server site, but a map service hosted on the server site can be used in ArcGIS Online with the map service URL. ArcGIS Online cannot host map services as ArcGIS Server, but can consume these services.

Currently, ArcGIS Online does not have a map service feature. As a workaround, publish a hosted tiled map service or vector tiled service from ArcMap or ArcGIS Pro. Refer to ArcGIS Online: Publish hosted tile layers from files for more information. This option allows sharing data outside of an organization whilst making the data non-editable.

Note:
Publishing a hosted service consumes credits based on the building and storage of the tile layer. To reduce credit consumption, build the tile package in ArcGIS Pro, and upload it to ArcGIS Online. Refer to ArcGIS Online: Build a tile package in ArcGIS Pro and upload to your organization for more information.