How restore from ASM to a new server

The REDO logs should be copied.

Shutdown doesn't trigger a logfile switch.

If you are changing the paths for the files, then using the "create controlfile reuse noresetlogs" method is the best. If you do move it without REDO logs, then archivelog current and then use the "create controlfile set resetlogs" method. In general though, moving the redo logs are best.

--EDIT--

Based on what you said, you are mounting these in the exact same path, then do your self a favor and just put all of the files, DATAFILES TEMPFILES CONTROLFILES REDOLOGS, etc in the exact same file path. Nothing else to do and it comes up fine.

-Richard

Try to "recover until SCN..." before doing "open resetlogs".

You'll need the online redo logs to perform this (not clear from your question whether you have them or not)