I doubt if the immortal person would find life boring if they loved any friends or relatives. Instead their life would be a frantic search for a way to make their friends and relatives also immortal - maybe everyone else in the world if they are altruistic enough - until it succeeds or all their friends and relatives die of old age, etc.
And if that search for giving immortality to their friends and relatives fails, the life of the immortal character may become like that of Flint in the episode "Requiem for Methuselah".
FLINT: And to conceal it. To live some portion of a life, to pretend to age and then move on before my nature was suspected.
I have married a hundred times, Captain. Selected, loved, cherished. Caressed a smoothness, inhaled a brief fragrance. Then age, death, the taste of dust. Do you understand?
I am assuming that Dubukay's immortal character is a man and not a woman or a child who had a $ 100.00 bill in their pocket.
1) If the immortal man is willing to tell the world about his new status.
What about selling some of "your" blood to medical researchers to study in hope that "your" blood transmits some of "your" immortality? How about selling small tissue samples?
I myself have had several small bits of skin taken to see if they were cancerous or not, and my two brothers have survived having larger parts of their internal organs removed because they were cancerous. People have donated spare kidneys to people needing new kidneys. So doctors certainly know how to remove small amounts of tissue and even entire organs safely for the person they are removed from.
What about selling some of "your" blood to desperate rich sick people in hope that "your" blood transmits some of "your" immortality? "You" should be able to get very high prices for it until and unless it soon proves not to work.
Presumably "you" are going to cut "your" hair and nails at the same rate as before. So maybe "you" should sell "your" hair and nail clippings as souvenirs or relics of the immortal man.
What about sperm donation? Make a deal with a sperm bank to sell sperm from the immortal man. People may hope that the sperm of the immortal man will make his children healthy immortal.
2) Jobs for the immortal man, whether or not he lets the world know about his status or keeps it secret as some answers suggest.
I don't know why you talk about an eternity of boredom. But maybe I don't get bored as easily as the "you" in your question. If "you" discovered "you" can't get injured or killed, "you" could plan on getting a series of highly dangerous jobs that are often considered to be exciting.
Of course some such jobs should have age limits, and organizations might not want to make exceptions for the Man Who Can't Be Killed merely because he thinks that he also doesn't age. So "you" should list those jobs in order of ascending age range and then plan on working in each one for a few years.
So "you" could become a soldier - probably in special forces if "you" can hack it - a policeman, a fireman, a miner, etc. etc. in the order of their upper age limits if they don't agree to make exceptions for "you" once it is proven that "you" don't age.
"You" could become a circus acrobat, a tightrope performer, a trapeze artist, a lion trainer.
"You" could become a movie stuntman and have a long career without worrying about getting crippled or killed. And it is possible that "you" might graduate to become an actor and maybe even a highly paid star. Some movie stars started as extras, stand ins, and stuntmen. And of course there is obviously a chance that a movie about "your" life will be made and "you" will get to star in it.
And someone planning a play or television show that they hope will run for many years or decades might want to hire a leading man who won't appear to age no matter how long it goes on for.
If "you" keep "your" condition secret, and are willing to commit crimes, "you" can blackmail one or more billionaires with fake deaths. If "you" have friends "you" trust "you' can arrange that billionaires or their family members have staged accidents in which something happens to "you" that nobody mortal could survive. But "your" confederates agree to hid the body and hush it up if they pay blackmail, and "you" and the accomplices split the money.
Accumulating a nest egg of a few hundred thousand or a few million dollars will be enough for an immortal to invest and become the richest person in the world in less than a century.
Since "you" had a $ 100.00 bill in your pocket to give to a homeless person, "you" don't seem like a total failure so far but reasonably successful in "your" career, whatever it is. So maybe "you" could continue in that career and try to be as successful as possible as fast as possible while saving as much of "your" earnings as possible and investing it as wisely as possible.
The best choice for investment would be funds that invest in a broad spectrum of companies so that returns and also risks are not as high as in some investments.
Some investments that are made each year should be marked to turn into cash in 50 years, others, in 60 years, others in 70 years, others in 80 years, others in 90 years, others in 100 years, others in 200 years, others in 300 years, others in 400 years, others in 500 years. If "you" make such investments for at least 100 years then "you" will turn those investments into cash beginning a mere fifty years after "you" start and continuing for centuries.
An investment growing at ten percent per year would multiply a thousand times in a little over 70 years. So every thousand dollars invested for a term of seventy plus years would become a million dollars at the end of that term (minus inflation, of course). So if one invests a thousand dollars for seventy plus years every year, at the end of the seventy plus years they will start getting a million dollars a year, minus inflation, year after year after year.
Every dollar invested for a time of 300 years at ten percent growth per year would become about a trillion dollars after 300 years, minus losses to inflation, but still many billions of dollars. So it would certainly be possible for an immortal person to arrange to someday have an income of many billions of dollars per year.
If an allegedly immortal character really is that immortal, then after their first ten thousand years of life, the decades (no more than a normal human life) of investing to get an income of a million dollars per year, and the centuries of investing to get an income of a billion dollars per year, and the centuries more of investing to get an income of a trillion dollars per year, and so on, will seem very short in retrospect, and it will seem to them as if they instantly became rich.
To them the decades and centuries of working and investing may seem no longer than the six years in elementary school seem to an normal old person looking back on their life. They will feel like they became super rich immediately after their childhood.