Is likelihood ratio test the only way to build hypothesis tests? Usually we can construct likelihood ratio for testing the Null hypothesis and alternative hypothesis:  The likelihood ratio test $ P( l(\beta_{1}) / l(\beta_{2}) ) < \alpha $ is the rejecting region for the null hypothesis. Then the inequality would reduce to a formula with sufficient statistics as the variable.
My question is : is likelihood ratio test the only way to build up hypothesis test? If not, what would be the alternatives? If there are multiple alternative methods, when would likelihood ratio test be preferred ?
 A: No, the likelihood ratio is not the only way to construct hypothesis tests, but it often is optimal.
In one flavour of the frequentist paradigm you can construct a hypothesis test from any arbitrary test statistic that can generate a p value ie a probability of observing the data, given the null hypothesis.  An alternative hypothesis does not need to be formally stated (other than "not null") and hence a likelihood ratio cannot be constructed.
Even when we do have a formal alternative hypothesis there are multiple ways of constructing tests, but the Neyman-Pearson lemma shows that in many situations the likelihood ratio will be the most powerful.  We are often seeking the most powerful test; or the "uniformly most powerful test" if the alternative hypothesis is composite (eg takes in multiple possible parameter values); or uniformly most powerful unbiased test if there is no clear uniformly most powerful.  So we often end up with a likelihood ratio test.
There are situations where likelihoods simply don't work - no density exists in the model for example.
The Bayesian paradigm gives an entirely different approach again, usually involving the calculation of a "Bayes factor" rather than a likelihood ratio.
