Probably the best and easiest way to kill ego is to practice devotion. It is because as long as you do anything else in someway or other it will indirectly affect your ego. But in devotion one considers himself as a mere servant of God and hence egoistic feelings will gradually decrease.
Ego is very difficult to overcome as it arises from the "I" consciousness. Hence it will always be there no matter what even though in a subtle amount as long as the jiva is alive. For example, when one will overcome desire, greed, etc. he will think I overcame desire, greed, anger etc. and the feeling of personal superiority or personal achievement will subconsciously boost the ego. So ego is very difficult and last to go. But in devotion, one practices to be humble and tolerant just like a grass and tree (trnad api su-nicena taror iva sahisnuna). A devotee feels himself as the lowest of all just like the grass, so the ego feeling is checked and he advances in spiritual practice.
Even though there can be many paths to get rid of negative thoughts, devotion is the only way to obtain the purity of mind. Only when tears of emotion flow from the eye that the mind is truly purified from negative feelings. So the scripture says:
kathaṁ vinā roma-harṣaṁ dravatā cetasā vinā
vinānandāśru-kalayā śudhyed bhaktyā vināśayaḥ [SB - 11.14.23]
If one’s hairs do not stand on end, how can the heart melt? And if the heart does not melt, how can tears of love flow from the eyes? If one does not cry in spiritual happiness, how can one render loving service to the Lord? And without such service, how can the consciousness be purified?
But apart from practicing devotion one common way to overcome ego is to know or associate with people who are better than you about which you have the egoistic feeling. For example, if one has ego about his wealth, then he should know or find the people who are richer than him. If one is proud of knowledge he should come in contact with people who are more intelligent than him. The simple knowledge that there are people out there who are better than oneself will certainly help a person to keep a check on his ego. To give a practical example, I have come across two personalities in my life who are just omniscient. One can tell you every single verse of any scripture you name and the other knows past, present and the future of anything and everything. Some years back I felt a bit proud about myself that I know so many things, but after coming in contact with them I can never ever have even an iota of ego no matter how much things I know.
Regarding fear, it is true that there are mantras that can help one overcome fear of certain things like snakes, ghosts, untimely death, etc. but I think those are better to be learnt from a teacher. Some mantras like the Mrutyunjaya mantra you already know which helps one overcome the fear of death.
If we analyse deeper, then we will find that fear and ego are the product of duality. Only then when the feeling and notion of others exist in the mind that any feelings of ego and fear can arise. But when through practice of devotion or practice of knowledge the non dual feeling that there exists nothing but only one supreme Brahman arises in the mind, the feelings of ego and fear automatically decrease. When there is nothing else but Me, whom to fear? When there is no one else but Me, who is superior or inferior? Thus the realization of non duality will also remove fear and ego. Hence, the scripture says in this regard:
yasmin sarvāṇi bhūtāny ātmaivābhūd vijānataḥ
tatra ko mohaḥ kaḥ śoka ekatvam anupaśyataḥ [Iso Up. -7]
One who always sees all living entities as spiritual sparks, in quality one with the Lord, becomes a true knower of things. What, then, can be illusion or anxiety for him?