Delphi работа со строками?

Проще всего так будет:

uses
Windows, Classes, UrlMon;

function LinesCount(): Integer;
var
lpBuffer : LPCSTR;
dwReserved: DWORD;
StringList: TStringList;
begin
Result := 0;
try
GetMem(lpBuffer, MAX_PATH);
ZeroMemory(lpBuffer, MAX_PATH);
GetTempPathA(MAX_PATH, lpBuffer);
GetTempFileNameA(lpBuffer, nil, 0, lpBuffer);
URLDownloadToFileA(nil, 'http://MySite.com/Stroki.txt', lpBuffer, dwReserved, nil);
StringList := TStringList.Create;
StringList.LoadFromFile(lpBuffer);
Result := StringList.Count;
DeleteFileA(lpBuffer);
finally
FreeMem(lpBuffer, MAX_PATH);
FreeAndNil(StringList);
end;
end;

procedure TForm1.Button1Click(Sender: TObject);
begin
Label1.Caption := IntToStr(LinesCount);
end;