A handful of Ammonium Sulfate to my Italian Cypress trees this Spring?

There are multiple reasons for using ammonium sulfate.

The ammonium is a source of nitrogen for soil bacteria to convert to nitrate, which is a plant fertilizer responsible for plant growth.  From http://www.gardeners.com/Fertilizer-Basics/5161,default,pg.html , "Nitrogen helps plants make the proteins they need to produce new tissues. In nature, nitrogen is often in short supply so plants have evolved to take up as much nitrogen as possible, even if it means not taking up other necessary elements. If too much nitrogen is available, the plant may grow abundant foliage but not produce fruit or flowers. Growth may actually be stunted because the plant isn't absorbing enough of the other elements it needs."

The sulfate combines with water to acidify the soil around your trees (something conifers prefer if your soil is too alkaline).  Think a very dilute version of sulfuric acid here.  The acidity makes some nutrients in the soil (iron and calcium, in particular) more readily available for your trees to use.  Iron is a component in chlorophyll, which makes carbohydrate (sugars) for the plants which are used as a source of energy as well as a structural component (cellulose).  From the same source as above, "Calcium is used by plants in cell membranes, at their growing points and to neutralize toxic materials. In addition, calcium improves soil structure and helps bind organic and inorganic particles together"  Also, sulfur is an element needed to form certain proteins.

It should be okay to continue your regular fertilizer with the ammonium sulfate.  Concerns of excess nitrogen are mostly for flowering plants (nitrogen promotes vegetative growth, not flower production or fruiting, but your cypress don't "flower" or produce vegetables) or the excess being carried away by runoff into other bodies of water promoting algal blooms (Arizona isn't what I'd consider a high-precipitation area, although you might be prone to flash floods - still, you might want to keep this in mind when watering if it becomes necessary).  You wouldn't necessarily have to start both at the same time, either.  I'd suggest starting the ammonium sulfate first to start the soil acidification, then follow up in a couple of weeks with the Tree & Shrub fertilizer.