Wow! Great improvement on the night already! Great!!
He is still young so I would keep your expectations low for now and understand that at this age sleep training is a more gradual process of teaching him that you will always return to him when he needs you, the trust bond that is being established by him sleeping on you for that first nap is not as 'bad' as it might seem. First decide if you are ready to get him into his cradle for this nap as you will need to expect it to be more work and more tiring for you than simply having him sleep on you, his nap will be disturbed and there may be fussing and crying. Some people choose to wait until baby is older before they begin sleep training, that's fine. I began slowly (and with no idea what I was doing) from birth really but I found BW and the EASY routine at roughly 4 wks and my LO was sleeping independently at around 8 wks (although I forget exactly when).
If you are ready then I would use shush/pat until he is asleep on you but right by the cradle, then continue to shush/pat as you put him down and continue to do it in the cradle until he is in deep sleep. If he fusses continue in the cradle, if he cries and is distressed then you pick up and continue shush/pat in arms until he is asleep again and continue in this way. He will get very very tired from the disturbed sleep (waking each time you try to lay him down) so the process of falling to sleep properly can take much longer, however that tiredness will eventually work on your side and he will fall to sleep and stay asleep. Just feeling the mattress beneath him begins to teach him he is safe there, with you there for reassurance. From there it's a gradual process to put down sooner and eventually to reduce the shush/pat in the cradle.
You might find these helpful
Shush-pat - How toTeaching Sleep to Newborns and Young InfantsSleep Training - the early daysYou can work on every nap or, if you feel this may be too much for you in one go then plan for one nap, the first nap of the day tends to be the one people work on first as LO should be quite well rested from their night sleep) but once you choose which nap then stick to that one, don't chop and change each day. Babies form habits quickly so if you are consistent with that first nap he will learn more quickly than if you keep changing the rules on him.
Let me know if you have any questions and how you get on.