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You can do this with some array image fiddling. That's always a bit tricky, and not very easy on the eyes. To understand this, I suggest that you add the intermediate array images to the map and inspect, to see what they contain.

var anomYearArray = anom
  .toArray() // Convert collection to 2D array image 
var anomArray = anomYearArray.arraySlice(1, 0, 1)
var anomYearSorted = anomYearArray
  .arraySort(anomArray) // Sort by anomaly
  

var rankArray = anomArray.not().not() // An array of 1
      .arrayAccum(0, ee.Reducer.sum()) // An array from 1 to number of anomalies
      .multiply(-1).add(anomArray.arrayLength(0)).add(1) // Reverse array
      
var anomYearRankArray = anomYearSorted.arrayCat(rankArray, 1)
var rankedAnom = ee.ImageCollection(
  ee.List.sequence(startYear, endYear)
    .map(function (year) {
      year = ee.Number(year)
      var yearMask = anomYearRankArray.arraySlice(1, 1, 2).eq(year)
      var yearImage = anomYearRankArray.arrayMask(yearMask)
        .arrayProject([1])
        .arrayFlatten([['anom', 'year', 'rank']])
        .set('year', year)
        .set('system:time_start', ee.Date.fromYMD(year, 1, 1))
      return yearImage
    })
)

var anom2020 = rankedAnom.filterMetadata('year', 'equals', 2020)
Map.addLayer(anom2020, {bands: 'rank', min: 1, max: endYear - startYear + 1}, 'rank 2020')

https://code.earthengine.google.com/82c841c8936fc562f6b17760876a9730

Here it is plugged into your complete script: https://code.earthengine.google.com/96066c8aea034d99bf4393eba8bf51f5