You'll need a new cable and housing (should come with the cable) to make it longer.
The bracket in the little plastic bag looks like it's for a 3 speed type setup. Your 5 speed cable housing should go right on the derailleur to hook up. Then add some zip ties or something fancier to hold the cable to the frame.
The size of the end nub in the shifter is what you need to have the same size. I think if you shift all the way forward (or back as Frankie mentioned) you can access the end, it slides out sideways and then the cable gets pulled out of the housing all the way. That's how the new one goes in, cable through the shifter housing then through the cable housing.
All the new cables I get are either shift cables or brake cables, the shift cables have smaller nubs on the ends. But I remember shifters from way back when that used larger nubs on the cables, so it's best to check the size of the old one for a match. If it's a large nub, a rear brake cable will probably work and be long enough.