Cypress is always a good choice for outdoor projects – it offers good stability and excellent decay resistance. It’s also reasonably easy to work with although it tends to be quite “knotty” (It is even referred to as Gopher Wood). You will need to take extra care when sawing, jointing, or routing to prevent chip-out on end-grain.
No special glue is required. Basic rule of thumb – outdoor use requires exterior grade glue. I prefer polyurethane glues for exterior work. If your local hardware store doesn’t stock it, you can get “Titebond” from Rockler.
“Titebond” also makes a less expensive wood glue and is great for all uses. It is their Titebond™ III Ultimate Wood Glue, also available from Rockler.
Tip: Take care when gluing — try not to get it on areas that are to be finished, Cypress does “show” glue stains more than some other varieties wood. Sometimes I even treat glue just like paint and mask off areas that I don’t want the glue.