Yes, it's a reasonable upgrade from an x2 5600+. The 5600+ scores 1148 on PassMark's CPUMark; the Phenom 9600 scores 2061 => so the upgrade would give you about an 80% increase in CPU "horsepower."
The most attractive thing about that particular upgrade is you can use your current motherboard.
If your goal is maximum performance, and you don't mind changing motherboards, you can get a substantially better improvement in performance by moving to an Intel Core-architecture based CPU. For example, the Q6600 (which costs LESS than a 9600B, and only a bit more than a 9600) scores 2680 on the PassMark benchmark, and the outstanding Q9450 (about 50% more) scores 3766. But to do this you'd have to upgrade to an Intel-based motherboard, such as the excellent Gigabyte P45-based GA-EP45-DS3L.
