You can try BCP in SQL server.

Yes there are lot of way we can import data from server to excel to server.

You can try BCP that is good idea to import data from table to excel...

you have the backup of that database. You have chance to get you table data from it.

restore the database in different name and get the information from that database table...

Try replication between the two server.

USE [master]

GO

ALTER DATABASE [AdventureWorks] SET RECOVERY FULL WITH NO_WAIT

GO

This is not a correct way but you can do at server at rest time.

Take the select * from table output in excel.

open the database choose the table Right Click...