Better way to write this code in functional manner using map and reduce?
Your “functional” approach is overly complicated. Notice that this is completely equivalent:

var result = arr.map(some_action).filter(other_action);


I.e. If you're only delegating to another function, you can specify that function directly. Also, an == true test is superfluous.

Your “procedural” variant is not equivalent, that would have to be:

var result = [];
arr.forEach(function(item){
    var changedItem = some_action(item);
    if (other_action(changedItem))
        result.push(changedItem);
});


Note that both variants have the same algorithmic complexity, and that the cost of iteration is likely negligible compared with the cost of some_action and other_action.

Functional programming does not mean unreadable code. Even if you're not just delegating to another function, you could improve formatting, e.g. to

var result = arr.map(function (item) {
  return some_action(item);
}).filter(function (item) {
  return other_action(item);
});