(1.) It is natural that he should be angry.

(2.) It is natural that he would be angry.

For the first sentence you gave, the phrase "<em>he should be angry</em>" means that "he" ought to be/should be expected to be angry in that situation. The first sentence implies that "his" anger is always justified in that specific situation. Importantly, this interpretation is that "he" (or anyone else for that matter) is expected to be or ought to be angry at something or someone. In other words, the word "should" implies "his" anger is justified in that situation (and that anyone else would be angry also if they were in his shoes). That is, of course, if "he" in fact gets angry at all.

For example:

For the second sentence you gave, the phrase <em>"he would be angry"</em><em> </em>has two possible interpretations. The first possible interpretation is the same as the one I listed above. This interpretation serves to specify that "he" (or anyone else for that matter) is expected to be or ought to be angry at something or someone. In other words, the word "should" implies "his" anger is justified in that situation. That is, of course, if "he" in fact gets angry at all.

The second possible interpretation is different in that "he" refers to only the individual that "would" be angry explicitly (and no one else implicitly if put into a similar situation). Moreover, this has no impact on whether or not that individual "should" be angry (in terms of an expectation as to what would usually make a person, on average, angry). Also, this interpretation refers to either someone who's angry all the time or someone who is already upset to begin with.

For example:

Are you asking for input or are you testing the readers??