Lion Tamarins are not possible near the city of SP and it would take more than one day to go and return to the areas where there is a very small possibility of seeing them (2 species in widely separated areas). However Parque Estadual da Serra da Cantareira has the following primates Brown Howler Monkey (easy to see), Brown Capuchin Monkey, Tufted-ear Marmoset and Masked Titi Monkey. Also, among the larger mammals, South American Coati, Ocelot, Tayra - the first relatively easy to see, the other two very difficult. PESC is easy to reach from Guarulhos airport but it is open only on weekends unless you have made prior arrangements with the park's administration or you hire a guide that has access to the park during weekdays. Birding is good but hampered by the late opening [ 8:30AM on weekends].
Campos do Jordão also mentioned by Xenospiza can also be done in a day. Best birding is along the Pedra do Bau road, a car is necessary. If you can manage some Portuguese you could negotiate with a taxi cab driver in Guarulhos (but not at the airport-only very expensive cabs from the airport co-op are available there). You can take the shuttle bus to one of the hotels in the city of Guarulhos and then negotiate with a taxi driver to take you to CJ, drive you around and return to the airport. It would be a bit more expensive than renting a car but you would have an experienced driver. Otherwise it might be worthwhile to hire a birding guide for a day. I believe that Rick Simpson, owner of the website mentioned by Xenospiza, can put you in contact with somebody in SP. It would be more expensive than renting a car or hiring a taxi driver but it would have other obvious advantages, specially if time is very short. But do notice time constraints: 2.5 hours to CJ, maybe another .5 hour to find Pedra do Bau road, then maybe another 3 hours to return, and one has spent 6 hours plus birding time.
I agree with Xenospiza that with just a day these two places would be the best birding options close to SP.