I have read somewhere that doing sadhana of mrityunjaya mantra can make a man immortal. If yes, what is the proper sadhana or practice or any other way to make a man immortal?
The proper practice to be followed in the use of the Mrityunjaya Mantra is described in this excerpt from the Rudra Samhita of the Shiva Purana. The sage Dadhichi once got into a fight with a king named Khshuva. Kshuva easily overpowered Dadhichi, so Dadhichi's ancestor Sukracharya (guru of the Asuras) came and magically healed his wounds, and then told him this:
Dear Dadicha, after worshipping Shiva the lord of everyone, I am going to tell you the highly potential Vedic mantra Mahamrityunjaya. We worship the three-eyed lord Shiva, The lord of the three worlds, the father of the three spheres, The Lord of the three gunas. Lord Shiva is the essence, the fragrance of the three Tattvas, three fires, of everything that is trichotomized, of the three world, of the three arms and of the trinity. He is the nourisher. In all living beings, everywhere, in the three Gunas, in the creation, in the sense organs, in the devas and Ganas, he is the essence as the fragrance in a flower. He is the lord of devas. O excellent brahmin of good rites, He is called the nourisher because it is from Him the supreme Purusha Siva that the Prakriti, the different Tattvas from Mahat to the different Indriyas, Vigm, Brahma, the sages, Indra and the devas derive their nourishment.
Worship that immortal deity Siva with sacred rites, penance, self-study of the Vedas, yogic practices, meditation, observance of truth and other means. You will be freed from the noose of Yama. The lord is the cause of both bondage and salvation. In my opinion this Mrtasaiijivani mantra is the most excellent of all. Repeat these mantras regularly remembering Siva with devotion. After Japa, Homa and recitation of the mantras observe fast, but you can drink water day and night. If the meditation is conducted in the presence of Siva there is no fear of death from anywhere. Nyasa and other ritualistic rites shall be observed. Siva shall be worshipped duly. Siva who is favourably disposed to his devotees shall be propitiated. I shall also mention the observance of meditation. It is after this meditation that the mantra shall be repeated as long as the purpose is realised due to Siva's power. [Emphasis added]
And here is what you need to meditate on before you say it:
I worship the three-eyed Lord Siva, the conqueror of death who is accompanied by(Parvati); who pours water on his head from two vessels held in his lotus-like hands, by means of the other pair of hands; who has placed the two hands with the pots on the lap; who usually holds in his hands the Rudraksha garland and a deer and whose body is rendered cool and wet by the nectar exuding from the moon worn qn the head.
I put in bold the instructions you need to follow. The consequences of all this is clear: "You will be freed from the noose of Yama." and "If the meditation is conducted in the presence of Siva there is no fear of death from anywhere."
And indeed, after Dadichi followed this procedure for a long time. Shiva appeared before him and granted him the following boons: "adamantine bones, impossibility of being killed and absence of distress."
On a side note, Dadichi didn't actually live forever, because he accepted death voluntarily for a good cause; when Indra was trying to kill the powerful demon Vritrasura, Dadichi gave up his own life so that his invincible bones could be fashioned into a weapon for Indra.
In any case, you should know that if you seek immortality as an end goal you probably won't get it. (Dadichi merely wanted invincibility as a means to teach a lesson to the arrogant king Kshuva.) As described in this excerpt from the Uttara Kanda of the Ramayana, Ravana and his brothers Kumbhakarna and Vibhishana were once engaging in Tapasya (deep meditation) for a very long time. Ravana even went so far as to cut off nine of his heads and made a move to cut off the tenth when Brahma appeared before the three brothers. Ravana asked for immortality, but he was denied it because he wasn't worthy. Vibhishana, on the other hand, asked to be good forever, and Brahma granted him immortality.
So if immortality is your goal, you may not get it, but by pursuing higher things you may get benefits like immortality and the like. It's akin to what Socrates said: "I tell you that virtue is not given by money, but that from virtue comes money and every other good of man, public as well as private." As Socrates says, it is Dharma or virtue that is the source of good things. And ultimately what should be pursued is not prolonging your life, but rather seeking an end to the entire cycle of life, death, and rebirth, namely Moksha.
There is no formula for immortality because nobody in the material world is immortal. Maha mrityunjaya mantra is recommended for those people who think their lives are in great danger like in case of some disease or some planetary influences. There is only one way to achieve iimmortality and it is to liberate oneself from this physical-material existence with the help of Hare Krishna mahamantra.