The part of the small intestine most likely to develop an ulcer from exposure to gastric juice is the

duodenum

Explanation:

Ulcers most commonly occur in the first part of the small intestine below the stomach known as the duodenum.  Usually, the lining of the intestine keeps the intestine from being hurt by stomach acid and digestive juices. If this protective layer breaks down, stomach acids can damage the walls of the intestine and cause an ulcer. A main cause of an ulcer is a bacteria you may have called Helicobacter pylori (H-pylori).