I am writing the answer as i have promised to do so to you in the chat room. But i think you'll be disappointed overall.
My answer is based on the Tantrasara. I have the 7th edition of the book.
[ If you want to know about the book or it's author Sri Krishnanada Agamvagis, then see this page. The author was an adept in Vedas and Tantras and he lived about 500 years ago in Bengal's Navadwip town. He traveled all over India and compiled his Tantrasara. It is the possibly the largest compendium work done on Tantras and Agamas. It is completely in Sanskrit, much later it was translated into Bengali by Rasikmohana Chattopaddhyaya. If you see this answer then Sir John Woodruffe is citing it, my answer there also citing from it. The book quotes from hundreds of Tantric and Agamic scriptures, some of which are not available at present as whole texts. Other works of this genre are the PrAnatoshini, SAradatilakam etc but TantrasAra is much more extensive and voluminous]
Because, the procedure is supposedly found in a Vaishnava Agama, so it is not advisable for anyone to chant this Mantra without initiation.
Moreover, the full procedures that are involved will appear to you as cumbersome, if you do not already know about how to perform NyAsas and other stuffs.
Anyways, first of all the correct Mantra in full is this:
उग्रं वीरं महाविष्णुं ज्वलन्तं सर्वतोमुखं । नृसिंहं भीषणं भद्रं
मृत्युमृत्युं नमाम्यहं ।।
So, the devotee should chant this and not the one which is found in the image in your question.
I have divided the answer into points (quoting from TantrasAra's pp 269):
1. RishiyAdi NyAsa:
All Mantras have limbs or Angas like DevatA, Chandah, Rishih etc. If someone chants a Mantra without having the knowledge of these limbs, then such a chanting is considered as Anga-HinA or devoid of limbs. Such chanting is not said to bear fruit.
So, first we need to know those limbs for this 32-letterd Mantra. It is said for this Mantra that:
Ugram viram mahavishnum jvalanatam... namAmahye.
Aya mantro mAyA putito bhavati tadA sarvArtha kAma phalapradah |
OR if the MAyA Bijam (Hrim) is added before and after the Mantra in the chant, then it fulfills all wishes of the devotee.
After doing PrAtakritya and Vishnu PujA, one chants this Mantra.
Shirasi brahmane rishaye namaha mukhe anushtupchandase namaha hride
nrisimhAya devatAyai namaha
From which we find that Brahma is the Rishih for the Mantra, Anushtupa the Chandah and Nrisimha the DevatA.
When we say "Shirasi brahmane rishaye namaha" we should touch the head with Tattva MudrA. Similarly when we say "mukhe anushtupchandase namaha" we should touch likewise our upper lips.. and the DevatA NyAsa is done in the heart.. These are called the RishiyAdi NyAsas.
This is done for purifying our body and hands, by invoking God-principle within our own body. So, we are making Nrisimha literally to come into our body. His hands are invoked in our hands and likewise for other body parts.
Because in Tantras it is said, that one can not worship God without becoming God himself.
These NyAsas are done in the following manner for this particular Mantra:
Ugram viram-angushtabhyAm namaha.. Mahavishnum-tarjanibhyam namaha..
Jvalanta sarvatamukham- madhyAbhyA vashat.... Nrishimham bhishana
anamikAbhyA hum... namAmyaha-astrAya phat...
Thereafter AngAdi NyAsas are done similarly.
3. Mantra NyAsa & DhyAnam
This NyAsa is done with Mantras like-
Right eye-bim namaha, left eye-ram namaha, face-mam namaha, right
armpit-hAm namaha, etc....
Then comes DhyAnam which is basically reciting some Sloka peculiar to the Deitry. This is said to bring the Deity immediately to the place where the chanting is done. Sometimes, it is not even advisable to do Japa without reciting the DhyAna Sloka first.
MAnikyAdri samaprabham nijaruchA samtrastar rakshoganam
jAnunyastakarAmbujam trinayanam ratnollasadbhushanam| ..(not the full DhyAnam)
The meaning of the DhyAnam is roughly this:
Nrisimha's body's complexion is like that of a mountain made of
precious jewels. And that glow is terrifying the demons. His hands are
resting on the thighs. He has three eyes and the body is decked with
clothing made with Ratnas. His two hands are holding the Shankha and
the Chakra. The face is terrifying because of the teeth and the
tongue, which is coming out and which looks like a flame of fire. Half
of his body is that of a lion and the other half is that of human.
After meditating on Lord Nrisimha thus, the devotee has to offer mental worship to him (MAnasopAchAra PujA).
Then he has to do Pitha PujA, Avarana PujA (that is done on the Nrisimha Yantra) and ShadAnga PujA like this:
MahAvishnum shirase swAhA|
Jvalantan sarvatomukham shikhAyai vaushat|
Nrisimham bhishanam kavachAya hum |... etc
This PujA must end with Dhupa, Visarjana etc..
Asya purashcharanam dvAtrimsallakshajapah |
TathAcha- Varnalaksham japenmantram tat sahasram ghritaplutaih|
PAyasAnnaih prajuhuyA dvidhivat pujitehanale ||
For PuraschArana of the Mantra one should do 32 lakhs of Japa and
thereafter 1/10-th of that number of oblations are to be offered in
the Homa. Oblation should be of Ghee made PAyasam.
The Brahma SamhitA have a Trailokya Vijaya Nrisimha Kavacha, which is comprised around this 32-lettered Mantra (this is found in TantrasAra's pp 591).
The Viniyogah of this Kavacha begins like:
TrailokyavijayasyApi kavachasya prajApatih rishih chandashcha gAyAtri
nrisimha devAta vibhuhu |
Khroum bijam me shirah pAtu chandravarno
Ugram viram mahAvishnum jvalantam sarvatomukham ||
Iti BrahmasamhitAyAm trailokyavijayam nAma nrisimha kavacham sampAptam
This made me to think that the Mantra is also found in the Brahma SamhitA, but the author does not mention the source.
He simply says this:
Athah Nrisimha Mantraha | Nivandhe- then he mentions the Mantra...