Would a 1996 dodge neon transmission work in 2001 efficiently?

No, a 1996 Dodge Neon transmission is highly unlikely to work efficiently, or at all, in a 2001 Dodge Neon. While both are Neons, there were significant changes to the platform and drivetrain components between 1996 and 2001. The transmissions themselves are likely different enough in terms of bellhousing bolt pattern, internal components, and even computer control to prevent installation, and even if somehow installed, they wouldn't function properly. You'd need a transmission specifically designed for the 2001 model year.