Why does the pronunciation of "U" vary in English?

It isn't so much a grammatical issue, as an historical one.

To make a very long story short, between the 1300s and 1700s, English underwent a change in pronunciation called the Great Vowel Shift. It was messy, and inconsistent. There are a few theories about why this happened, but there isn't any grand consensus.

Unfortunately for you and every other English language student, the standardization of spelling happened right in the middle, between the 1500s and 1600s. So, many of the spellings were created to reflect older pronunciations, which are no longer used today.

In other cases, the different pronunciations of the letter reflect the language of origin. English is a basically a mishmash of Anglo-Saxon, Norman French, and Latin, with bits of Dutch and Old Norse and others thrown in for fun, based on a long history of conquerors taking over the British Isles.

As it happens, none of the examples you've given in your question are of actual English origin: umbrella is Latin via Italian, and use, utensil, study and student are Latin via French. Other words like up and under are from Old English, while ugly comes from Old Norse. Here, you can already see some patterns, that the words of Latin origin tend to have one sound (use, utensil, student), while the words of Old English / Germanic origin tend to another (up, under, ugly), but even that isn't consistent (study, umbrella). Roughly speaking, these correspond to IPA /u/ and /ʌ/ respectively.

Other examples: great big clusters of vowels tend to point towards a French origin, like oeuvre and bouquet. Clusters of mostly silent consonants tend to point towards very old words of a Germanic origin (brought, knight), where one or more consonants have been elided over centuries of usage.

tl;dr: There are no hard and fast rules for determining the pronunciation of English words directly from spelling. In fact, even native English speakers will occasionally trip over the pronunciation of a particular word, particularly when it is seen written many times before hearing it spoken (from personal experience).

There have been attempts to resolve this issue, which can be fascinating to read about, but as of yet, none have stuck.

In cases of doubt, Wiktionary does a wonderful job of showing pronunciations in IPA format for British and American English, and there is usually at least one audio sample to listen to.

As a commenter notes, the issue is that English has more vowel sounds than it does letters for vowels. A similar issue was noted by an anonymous scholar of the twelfth century trying to devise an alphabet for Icelandic (which also has many vowel sounds). The scholar discusses in a relevant part why he decided to add some new vowel letters to the five from the Latin alphabet:

Now it may well happen that some one will speak up in this way: 'I can read the Danish language perfectly well, even though it be written with the proper Latin letters. I can make out what it says, even when some of the letters in what I read can not be pronounced correctly ... To that I say: it is not the virtue of the letters that enables you to read and to make out the meaning where the letters are unclear. That is rather YOUR virtue, and it is not to be expected that I also, or any one else like me, if such there be, shall be able to read well and to make out which path to take where more than one course is possible because it is written one way, but not clearly, and we then have to guess, as you claim you can do so well.

Haugen (1950) "First Grammatical Treatise"

English readers effectively become good at "guessing" how to pronounce a word, since we don't have enough vowels to write with.