[SalesForce] OnClick Javascript Ajax Toolkit, Custom button update a field

update_Case is a variable at the point where you're referencing it in the if condition so you don't need the quotes nor the merge syntax.

A couple of other points, you should always declare JavaScript variables with the var keyword and I would suggest using the triple-equals operator in the if expression to avoid type coercion:
if (update_Case.pl_ReasonForAssignment__c === '') {

{!REQUIRESCRIPT("/soap/ajax/30.0/connection.js")}

var updateRecord = new Array();
var myquery = "SELECT Id, txFaultRequestInfo__c, pl_ReasonForAssignment__c FROM Case WHERE Id = '{!Case.Id}' limit 1";

var result = sforce.connection.query(myquery);
var records = result.getArray("records");

if (records[0]) {
    var update_Case = records[0];

    // a little debugging
    alert(update_Case.pl_ReasonForAssignment__c);

    if (update_Case.pl_ReasonForAssignment__c == '') {
        alert("Picklist value is not set!");

    } else {
        update_Case.OwnerId = "00G20000000jeqC"; /* New Value */
        //update_Case.OwnerId = "00G20000000jdLD"; /* Older Value */

        updateRecord.push(update_Case);
    }
}

result = sforce.connection.update(updateRecord);
parent.location.href = parent.location.href;

I would just use a controller property to control the visibility. You also need to use the rerender property for the commandButton. Something like this

VF Page

<apex:commandButton action="{!show}" value="Show" id="cb" rerender="myTable"/>

<apex:pageblocktable rendered="{!showTable}" value="{!Departments}" var="dept_ref" id="myTable">

     <apex:column value="{!dept_ref.Subject__c}"/>
     <apex:column value="{!dept_ref.Syllabus__c}"/>
     <apex:column value="{!dept_ref.Employees__c}" headervalue="Emp count"/>
     <apex:column headervalue="Name" value="{!dept_ref.Name}"/>

</apex:pageblocktable>

APEX

public class myController {

    public boolean showTable      {get;set;}

    public myController() {
         showTable = false;
    }

    public void show() {
         showTable = true;
    } 
}