The Abschlussmitteilung is a final letter that BAMF will send to the local immigration authority (Ausländerbehörde) where you live. It confirms the status that BAMF has decided to give you. If you have already received your letter (Bescheid) from BAMF, then it should not take much longer for them to receive this Abschlussmitteilung. It's possible that if they currently have many applicants from the Ukraine, it could take longer for them to contact you.
You can already send an application for a residence permit to the Ausländerbehörde with a copy of the BAMF Bescheid. This should hopefully speed up the process.
If you have been receiving financial support from the Sozialamt up to now, you should make an application to the local Jobcenter in order to continue to receive financial support. They can begin to process your application and once you have either a residence permit or Fiktionsbescheinigung (a document from the Ausländerbehörde to prove you have applied for a residence permit) they can provide you with the money you need.
Let us know if you have any other questions.