How to split string between 2 patterns in javascript

Try a regular expression.

Here's a good primer on how they work: https://www.codepicky.com/regex/

/_(\d{3,4})\.json$/

What is happening with this pattern?

1. The beginning and ending / are simply bookends defining a pattern
2. The _ literal will match the underscore that precedes the digits
3. (\d{3,4}) defines a "capture group" that matches exactly 3 or 4 consecutive numeric digits. This is handy because it lets us extract the digits you want separately from the overall pattern.
4. \.json$ matches the string .json (you have to escape the period with a slash because it is a special regex character) and the $ enforces it being at the end of the string

Example:

let result1 = "test_halloween Party 10 AM - 12:30 PM party_560.json".match(/_(\d{3,4})\.json$/);
// result1[1] === 560

let result2 = "Kaulampur 1110 reva_2018  RR_999 Roadrover_4987.json".match(/_(\d{3,4})\.json$/);
// result2[1] === 4987

let result3 = "this string will not match".match(/_(\d{3,4})\.json$/);
// result === null

Regular expressions are extremely versatile, precise, and fast. Take a look at this benchmark comparing it to a string index-finding alternative: http://jsben.ch/lbfUt

I'd solve it like this (slower than pre-compiled regex):

function myFunc(s) {
    let i = s.lastIndexOf("_");
    let j = s.indexOf(".", i);
    return s.substring(i+1, j);
}

console.log(
    myFunc("test_halloween Party 10 AM - 12:30 PM party_560.json"),
    myFunc("Kaulampur 1110 reva_2018  RR_999 Roadrover_4987.json")
);

Anyone interested in the hand-coded DFA mentioned in comments:

function myFunc(s) {
  const MAX = 10;
  t = s.substr(-MAX);
  for (let i=0; i<MAX; i++) {
    let z = "";
    if (t[i] === "_") {
      i++;
      if (isd( t[i] )) {
        z += t[i];
        i++;
        if (isd( t[i] )) {
          z += t[i];
          i++;
          if (isd( t[i] )) {
            z += t[i];
            i++;
            const IS_DOT = 1;
            const IS_DIGIT = 2;
            let x = (t[i] === ".")
              ? IS_DOT 
              : (isd(t[i]))
                ? IS_DIGIT
                : 0;
            OUT:
            while (true) {
              switch (x) {
              case IS_DOT:
                i++;
                if (t.substring(i) === "json") {
                  return z;
                }
                break;
              case IS_DIGIT:
                z += t[i];
                i++;
                x = IS_DOT;
                break;
              default:
                break OUT;
              }
            }
          }
        }
      }
    }
  }
  return null;
}

function isd(c) {
  let x = c.charAt(0);
  return (x >= "0" && x <= "9");
}

console.log(
    [
        "_asnothusntaoeu_2405.json",
        "_asnothusntaoeu_105.json",
        "_asnothusntaoeu_5.json",
        "_asnothusntaoeu.json",
        "_asnothusntaoeu_5json",
        "_asnothusntaoeu_5.jso",
        "_asnothusntaoeu_105.json"
    ].map(s => myFunc(s))
);

try this var res1 = /([0-9]+)\.json$/.exec(str1)[1];

This seems like a textbook case of when you'd just want to use a Regular Expression. Something like:

// Select all things of the form "_<numbers>.json" from 
// the string, and parse out <numbers> as a match.
var MyRegEx = /_(\d+)\.json/i;

var str1 = "test_halloween Party 10 AM - 12:30 PM party_560.json";
var res1 = MyRegEx.exec(str1)[1];

var str2 = "Kaulampur 1110 reva_2018  RR_999 Roadrover_4987.json";
var res2 = MyRegEx.exec(str2)[1];

console.log(res1);    
console.log(res2);

That should do the trick.