How to sort an arraylist on date?

Basically, don't try and work with the file as lines of text; separate that away, so that you have one piece of code which parses that text into typed records, and then process those upstream when you only need to deal with typed data.

For example (and here I'm assuming that the file is tab-delimited, but you could change it to be column indexed instead easily enough); look at how little work my Main method needs to do to work with the data:

using System;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Linq;
static class Program
{
    static void Main()
    {
        foreach (var item in ReadFile("my.txt").OrderBy(x => x.Joined))
        {
            Console.WriteLine(item.Names);
        }
    }
    static readonly char[] tab = { '\t' };
    class Foo
    {
        public string Names { get; set; }
        public int Age { get; set; }
        public string Designation { get; set; }
        public DateTime Joined { get; set; }
    }
    static IEnumerable<Foo> ReadFile(string path)
    {
        using (var reader = File.OpenText(path))
        {
            // skip the first line (headers), or exit
            if (reader.ReadLine() == null) yield break;

            // read each line
            string line;
            var culture = CultureInfo.InvariantCulture;
            while ((line = reader.ReadLine()) != null)
            {
                var parts = line.Split(tab);
                yield return new Foo
                {
                    Names = parts[0],
                    Age = int.Parse(parts[1], culture),
                    Designation = parts[2],
                    Joined = DateTime.Parse(parts[3], culture)
                };
            }
        }
    }
}

And here's a version (not quite as elegant, but working) that works on .NET 2.0 (and probably on .NET 1.1) using only ISO-1 language features; personally I think it would be silly to use .NET 1.1, and if you are using .NET 2.0, then List<T> would be vastly preferable to ArrayList. But this is "worst case":

using System;
using System.Collections;
using System.Globalization;
using System.IO;
class Program
{
    static void Main()
    {
        ArrayList items = ReadFile("my.txt");
        items.Sort(FooByDateComparer.Default);
        foreach (Foo item in items)
        {
            Console.WriteLine(item.Names);
        }
    }
    class FooByDateComparer : IComparer
    {
        public static readonly FooByDateComparer Default
            = new FooByDateComparer();
        private FooByDateComparer() { }
        public int Compare(object x, object y)
        {
            return ((Foo)x).Joined.CompareTo(((Foo)y).Joined);
        }
    }

    static readonly char[] tab = { '\t' };
    class Foo
    {
        private string names, designation;
        private int age;
        private DateTime joined;
        public string Names { get { return names; } set { names = value; } }
        public int Age { get { return age; } set { age = value; } }
        public string Designation { get { return designation; } set { designation = value; } }
        public DateTime Joined { get { return joined; } set { joined = value; } }
    }
    static ArrayList ReadFile(string path)
    {
        ArrayList items = new ArrayList();
        using (StreamReader reader = File.OpenText(path))
        {
            // skip the first line (headers), or exit
            if (reader.ReadLine() == null) return items;

            // read each line
            string line;
            CultureInfo culture = CultureInfo.InvariantCulture;
            while ((line = reader.ReadLine()) != null)
            {
                string[] parts = line.Split(tab);
                Foo foo = new Foo();
                foo.Names = parts[0];
                foo.Age = int.Parse(parts[1], culture);
                foo.Designation = parts[2];
                foo.Joined = DateTime.Parse(parts[3], culture);
                items.Add(foo);
            }
        }
        return items;
    }
}

I'm not sure why you'd want to retrieve just the dates. You'd probably be better reading your data into Tuples first. Something like

List<Tuple<string, int, string, DateTime>> items. 

Then you can sort them by items.Item4, which will be the date.

You can use LINQ and split the line according to tabs to only retrieve the date and order them through a conversion to date.

while ((linevalue = filereader.ReadLine()) != null)
    {
          items.Add(linevalue.Split('\t').Last());
    }
    filereader.Close();
    items.OrderBy(i => DateTime.Parse(i));
    foreach(var item in items)
    {
        Console.WriteLine(item);
    }

get the desired values in Array from the file...

public class DateComparer : IComparer {
  public int Compare(DateTime x, DateTime y) {
    if(x.Date > y.Date)
         return 1;
    if(x.Date < y.Date)
         return -1;
    else
         return 0;
  }
}
list.Sort(new DateComparer());