It sounds like you both have made your mistakes. Sometimes when a relationship follows mistakes it is helpful to get assistance from a professional. That is only however if you want this to work. And by working I mean that you accept the mistakes he has made especially the online affair. You would have to resolve feeling betrayed and he would have to resolve his feelings about your one night stand as well. You both have to let this go for the relationship to work. It cannot work if he is going to revisit this issue every day. He can't make you miserable because he can't let it go.
If this partnership doesn't get more peaceful then the prospects are not good. You both have to be committed. See a couples counselor to see if this can be fixed. There has to be trust and consideration or counseling isn't going to be productive. Consult with him to see how dedicated he is. You can't be the only one that is going to compromise and put in the work. He may not be the type to let things go. That is not a good sign.
I would approach him about this and see what he is willing to do. Key points in a relationship have to be developed - trust, communication and mutual respect. If you are the only one motivated then it may be time to move on.
Please press accept; this is the only way I am compensated