Learn English – 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.