But how would field lines look like if we put a metal ring in the
center of that container? Will the picture be similar to this?
The electric field lines outside the ring would approximately look like your picture
And will there be any field lines inside the ring?
No. In the diagram below I am assuming the direction of the external electric field lines follows the normal convention that it is the direction of the force that a positive charge would experience if placed in the field. Now the free electrons in the conductive ring will migrate in the direction where they are attracted by the force of the external field, leaving a net positive charge on the other side of the ring, as shown in the diagram (sorry if the + and - charges are hard to see. Just blow up the diagram).
This results in an electric field in the ring itself that is equal and opposite the external electric field, in effect cancelling the external field. End result is no electric field inside the ring.
Hope this helps.