On an Exchange 2013 server I can create an smtpReceiveAgent transport agent to perform the task shown on this page, which is this c# code (VB code is also available on the webpage): -

using System;
using System.Collections.Generic;
using System.Text;
using Microsoft.Exchange.Data.Transport;
using Microsoft.Exchange.Data.Transport.Smtp;

namespace MyAgents
{
    public sealed class MyAgentFactory : SmtpReceiveAgentFactory
    {
        public override SmtpReceiveAgent CreateAgent(SmtpServer server)
        {
            return new MyAgent();
        }
    }
    public class MyAgent : SmtpReceiveAgent
    {
        public MyAgent()
        {
            this.OnEndOfData += new EndOfDataEventHandler(MyEndOfDataHandler);
        }
        private void MyEndOfDataHandler (ReceiveMessageEventSource source, EndOfDataEventArgs e)

        {
            // The following line appends text to the subject of the message that caused the event.
            e.MailItem.Message.Subject += " - this text appended by MyAgent";
        }
    }
}

How can I do something similar on my Office 365 Exchange implementation?

Thanks, Paul.

You can't do anything on the Office365 servers outside of Transport Rules and DLP which is the best way to solve the problem your talking about.  How third parties are implementing solutions is though Connectors https://technet.microsoft.com/en-us/library/ms.exch.eac.connectorselection(v=exchg.150).aspx  eg https://www.exclaimer.com.au/exclaimer-cloud/signatures-for-office-365/technical . (but this then requires you have your own servers to the process the messages).

Cheers
Glen