SQL Server – Backup and Restore Large Database with Limited Disk Space

But my question here would be how can i predict or calculate what size of drive would be require to accommodate the backups

From Books Online Document. If you see the last part it says that

For compressed backups, the size of the final backup file depends on how compressible the data is, and this is unknown before the backup operation finishes. Therefore, by default, when backing up a database using compression, the Database Engine uses a pre-allocation algorithm for the backup file. This algorithm pre-allocates a predefined percentage of the size of the database for the backup file. If more space is needed during the backup operation, the Database Engine grows the file. If the final size is less than the allocated space, at the end of the backup operation, the Database Engine shrinks the file to the actual final size of the backup.

To allow the backup file to grow only as needed to reach its final size, use trace flag 3042. Trace flag 3042 causes the backup operation to bypass the default backup compression pre-allocation algorithm. This trace flag is useful if you need to save on space by allocating only the actual size required for the compressed backup. However, using this trace flag might cause a slight performance penalty (a possible increase in the duration of the backup operation).

When you start compressed backup you would see some size of backup file created on the drive but this would not be the correct size during backup operation it can grow and final size would increase. You can initiate a backup with compression and see what is the size then tentatively multiply it by number of days you want to keep it plus few more space. This would again give you tentative size of backup drive. I would also say it would be safer to more space to backup drive

Your options are pretty limited.

1. Get enough disk space to restore this database. When restoring a backup, SQL Server needs to create the data files, even if the majority of the space is unused. After it's restored, you can shrink the files and move them, but ultimately you'll need that 1.6TB of disk space temporarily in order to perform the restore. If you are using a temporary drive just for this restore, it doesn't have to be good/fast storage--it just has to be storage.
2. Get a new backup. You can have your client shrink the database, then take a new backup and send you a copy of the post-shrink database.