According to EU rules, you should receive a residence permit and you can use that to travel for up to 90 days within 180 days to other Schengen countries. With the same permit, you can leave the EU to another third country – for example Ukraine – and re-enter the EU again.
If you have not yet received your residence permit, you may receive national papers or permits. This is different from one EU country to another. Certain papers or permits – those listed in Annex 22 to the Schengen Handbook – allow you to travel as if you would have a residence permit. However, this is not the case for all national papers/document issued by Schengen countries. Please check with the national authorities of the country where you are registered for temporary protection if the papers you receive allow you to travel.
If this is not the case, but you need to travel to Ukraine and come back, the only viable solution would be for you, when you come back to the EU, to enter directly – by land, air or sea- to the EU country where you are registered for temporary protection.
If you need to travel to another Schengen country you can do so for up to 90 days within 180 days with your residence permit or a national paper/document which is listed in Annex 22 (and therefore has the same effect as a residence permit). If you don´t have such document you are advised not to travel for the time being to other Schengen countries, since your stay there might be considered illegal.
!!Attention!! Residence permits issued by Romania, Bulgaria, Croatia, Cyprus and Ireland don´t have Schengen wide effect and don´t give you the right to travel to other Schengen States. Residence permits issued by States that are in the Schengen area without internal border controls are, however, recognized by Romania, Bulgaria, Croatia and Cyprus and give a right to travel to these countries.