There are many methods to check file write permission on Asp.net shared hosting. Quickest and easiest method for this purpose is creating a file and deleting that for this purpose.

Here is sample method to do this

/// <summary>
/// Helper method to quickly check if directory or file is writable
/// </summary>
/// <param name="url">Physical path</param>
/// <param name="file">Optional File name</param>
/// <returns>True if application can write file/directory</returns>
public static bool CanWrite(string url, string file = "")
{
    var dir = HttpContext.Current.Server.MapPath(url);

    if (dir != null && Directory.Exists(dir))
    {
        if (string.IsNullOrEmpty(file)) 
            file = string.Format("test{0}.txt", DateTime.Now.ToString("ddmmhhssss"));

        try
        {
            var p = Path.Combine(dir, file);

            using (var fs = new FileStream(p, FileMode.Create))
            using (var sw = new StreamWriter(fs))
            {
                sw.WriteLine(@"test");
            }

            File.Delete(p);
            return true;
        }
        catch (Exception)
        {
            return false;
        }
    }
    return false;
}