I followed following things,

Created a new .NET 8 ASP Core Webapp.
Added Microsoft.AspNetCore.SystemWebAdapters dependency to it.

Program.cs

using OldHandlers;

var builder = WebApplication.CreateBuilder(args);

// Add services to the container.
builder.Services.AddRazorPages();
builder.Services.AddSystemWebAdapters(); // Register SystemWebAdapters

var app = builder.Build();

// Configure the HTTP request pipeline.
if (!app.Environment.IsDevelopment())
{
    app.UseExceptionHandler("/Error");
    app.UseHsts();
}

app.UseHttpsRedirection();
app.UseStaticFiles();
app.UseRouting();

// Register SystemWebAdapters middleware
app.UseSystemWebAdapters();

// Define endpoints
app.UseEndpoints(endpoints =>
{
    endpoints.MapGet("/myhandler", async context =>
    {
        var adapter = new AspNetCoreHttpContextAdapter(context);
        var handler = new MyHandler();
        handler.ProcessRequest(adapter);
    });


});

app.UseAuthorization();
app.MapRazorPages();

app.Run();

AND,

using Microsoft.AspNetCore.Http;
using Microsoft.AspNetCore.SystemWebAdapters;
using System.Threading.Tasks;
using OldHandlers;

public class AspNetCoreHttpContextAdapter : IHttpContextAdapter
{
    private readonly HttpContext _context;

    public AspNetCoreHttpContextAdapter(HttpContext context)
    {
        _context = context;
    }

    public string ResponseContentType
    {
        get => _context.Response.ContentType;
        set => _context.Response.ContentType = value;
    }

    public void WriteResponse(string content)
    {
        _context.Response.WriteAsync(content);
    }

    public async Task<string> ReadRequestBodyAsync()
    {
        using (StreamReader reader = new StreamReader(_context.Request.Body))
        {
            return await reader.ReadToEndAsync();
        }
    }

    public void SetStatusCode(int statusCode)
    {
        _context.Response.StatusCode = statusCode;
    }

    public object GetRouteData(string key)
    {
        return _context.Items[key];
    }
}

In my Old website, created a new handler,

 public class MyHttpHandler : IHttpHandler, IRequiresSessionState
 {
     public void ProcessRequest(HttpContext context)
     {
         var adapter = new SystemWebHttpContextAdapter(new HttpContextWrapper(context));
         var handler = new MyHandler();
         handler.ProcessRequest(adapter);
     }

     public bool IsReusable => false;
 }

AND,

using System.IO;
using System.Threading.Tasks;
using System.Web;
using OldHandlers;

public class SystemWebHttpContextAdapter : IHttpContextAdapter
{
    private readonly HttpContextBase _context;

    public SystemWebHttpContextAdapter(HttpContextBase context)
    {
        _context = context;
    }

    public string ResponseContentType
    {
        get => _context.Response.ContentType;
        set => _context.Response.ContentType = value;
    }

    public void WriteResponse(string content)
    {
        _context.Response.Write(content);
    }

    public async Task<string> ReadRequestBodyAsync()
    {
        using (StreamReader reader = new StreamReader(_context.Request.InputStream))
        {
            return await reader.ReadToEndAsync();
        }
    }

    public void SetStatusCode(int statusCode)
    {
        _context.Response.StatusCode = statusCode;
    }

    public object GetRouteData(string key)
    {
        return _context.Items[key];
    }
}

AND created a .NET Standard 2.0 library to have the concrete handler implementation,

namespace OldHandlers
{

    public class MyHandler
    {
        public void ProcessRequest(IHttpContextAdapter context)
        {
            context.ResponseContentType = "text/plain";
            context.WriteResponse("Hello, World!");
        }
    }
}

using System;
using System.Collections.Generic;
using System.Text;
using System.Threading.Tasks;

namespace OldHandlers
{
    public interface IHttpContextAdapter
    {
        string ResponseContentType { get; set; }
        void WriteResponse(string content);
        Task<string> ReadRequestBodyAsync();
        void SetStatusCode(int statusCode);
        object GetRouteData(string key);
    }

}

What I don’t understand is that, if I use Microsoft.AspNetCore.SystemWebAdapters https://www.nuget.org/packages/Microsoft.AspNetCore.SystemWebAdapters then do I need to create my adapter classes, AspNetCoreHttpContextAdapter and SystemWebHttpContextAdapter?