The soap that worked the best for a friend of mine who struggled with cystic acne was 50% hazelnut, 25% coconut and 25% palm. I think any high oleic oil will work in place of hazelnut but I haven't tried to replace it. I wouldn't use neem oil myself for blackheads. The most you will get from a soap is to find a combination of oils that won't make the problem worse.
If you are battling mostly blackheads, consider visiting a dermatologist and getting a prescription for a product with high concentration of salicylic acid. It would be similar to this one:
http://www.paulaschoice.com/shop/skin-care-categories/targeted-treatments/_/Resist-BHA-9. The active ingredient is 9% salicylic acid dissolved in propylene glycol.
I make a similar solution for my own use. I only use it occasionally (once a week or less) around my nose to help reduce blackheads. I wouldn't use it on a regular basis without the supervision of a dermatologist (particularly for a growing teen) even though it is sold in the US without a prescription like Proactive.
You can read more about it here:
https://www.nlm.nih.gov/medlineplus/druginfo/meds/a607072.html