Are there alternatives to the Bayesian update rule? Are there any other methods to update my belief in a hypothesis aside from the Bayesian update rule?
 A: There are some alternatives, in fact, but they rely on using non-probabilistic methods. (The uniqueness of Bayes Law is implied by the uniqueness of a single probability measure, and the definition of joint probability - see this other answer for details)
Dempster Shafer theory is an alternative, as are more complex formalisms such as DSMT. Similarly, Imprecise probabilities can do similar things - but they still use Bayes' law. Fuzzy Logic and other formalisms may also be of interest.
A: I'll add another perspective.  In E. T. Jaynes incredible book Probability Theory: The Logic Of Science, he gives a rigorous treatment of an extension of Aristotelian logic to degrees of belief (Jaynes was a baysean to the core).  That is, he explores a probability theory so that $P = 1$ and $P = 0$ correspond to True and False in classical first order logic.
He outlines a few desirable properties that such an extension must have:

I. Degrees of Plausibility are represented by real numbers.
II. Qualitative Correspondence with common sense.
IIIa. If a conclusion can be reasoned out in more than one way, then
  every possible way must lead to the same result.
IIIb. Always take into account all of the evidence
  relevant to a question. Do not arbitrarily ignore some of
  the information, basing its conclusions only on what remains.
  In other words, the theory is completely non-ideological.
IIIc. Always represent equivalent states of knowledge by
  equivalent plausibility assignments. That is, if in two problems
  the state of knowledge is the same (except perhaps for
  the labeling of the propositions), then it must assign the same
  plausibilities in both.

all of these statements are given rigorous interpretations in the book.
Then Jaynes gives a deeply fascinating and rigorous mathematical demonstration that Bayesian reasoning is the unique extension of logic to degrees of belief that satisfies these requirements.  So in the sense of Jaynes, the answer is actually no, if you want your theory to be compatible with Aristotle, Bayes is the only way.
A: To my knowledge, if you assign a probability to your belief, the bayesian updating rule is the only way to act upon new datas in a consistent manner in line with probabilities.
You might have two reasons to leave the bayesian framework :    


*

*You don't want to assign probabilities to a belief.    

*You don't have (or don't want to specify) an alternative belief.


One (amongst other) alternative to Bayesian Inference is the framework of Null Hypothesis Statistical Testing (NHST) framework to eventually reject your hypothesis. One could argue that rejecting a belief is a hard form of updating. You do not mention in your question an alternative hypothesis. 
If your belief has some degree of freedom, this is a special case. There is no straight answer of how to judge which model is the best as different criteria coexist. There is a bayesian way to act (Bayesian Information Criterion), but also others (the field is known as model selection). I don't know if and how a belief can be updated in this case.
