Hola Don

The verb arrepentir is always used pronominally, i.e. using the reflexive pronoun, and you can't have an object, therefore you just say: "no te arrepentirás" - it's not possible to say for example "no te lo arrepentirás" (incorrect with lo).

You can also use it with "de" as in "arrepentirse de hacer algo/ de algo". For example:

No te arrepentirás de venir a la fiesta.

You won't regret coming to the party.

No te arrepentirás de tu sinceridad. 

You won't regret your honesty. (=being honest).

With lamentar, you can use it as you said, as a transitive verb with a direct object: lamentar algo 

"No lo lamentarás" (regret)

This has exactly the same meaning as "No te arrepentirás" (regret)

However, "lamentarse de algo" is also another way to use this verb, but the connotation is different. It is used when you want to express a "complaint" (also feeling sorry about something): 

Se estaba lamentando de su mala suerte.

He was complaining / feeling sorry about his bad luck.

Here's the entry for lamentar / lamentarse in the Collins dictionary where you can see different uses and slightly different meanings (scroll down to see it all). 

Saludos