Major revision has been made, in response to By137's comment, as follows:
To begin with, consider the origin of もったいぶらず. Where does it come from? もったいぶらず, or equivalently もったいぶらないで, is an adverb which means doing something in a not もったいぶる way. もったいぶる is a verb which means that someone is unwilling to do something because it is もったいない．
Next, consider the meaning of adjective もったいない. When do we think that doing something is もったいない? First, it is もったいない when valuable resources, such as money, time, or effort, are spent in a not useful/efficient/effective way. A typical usage is as follows:
It's a waste of time.
(This implies that there can be a more useful way of spending time.)
It is also もったいない when something is being done in favor of someone who does not deserve it. In other words, it is もったいない when it is not worthy enough to spend some valuable resources in favor of someone. A typical usage is as follows:
You don't deserve it.
(This implies that there can be someone else who really deserves it.)
In both usages, the key to understand the meaning of もったいない is a sense of values/worthiness. Of course, everyone has a different sense of values, and whether to think that something is もったいない depends on who you are. In other words, when to say もったいない is a subjective matter. We shall return to this point at the end of the answer.
Now, go back to the main question. What does もったいぶらず mean? We can infer the meaning from its origin and from もったいない: Doing something もったいぶらず means doing something without 'hesitation that is caused when one considers whether it is worthy enough to do.' For example, in the manga, the boxer said:
I'm gonna spend all (the money) I have right now (without considering how valuable the money is).
(The boxer is very willing to pay all his signing bonus in order to drink in a hostess bar.)
Then, think about what a もったいぶらない way of boxing is. After saying (1), the boxer added that:
(2) それが俺の武器に繋がるんでねェ。拳に表れるんだよォ ――――生き方スタイルってヤツは。
That will lead to my strong point. In my fist ―is my way of life (which is もったいぶらない).
(The boxer is trying to say that his もったいぶらない characteristic is his strong point and that he もったいぶらない not only in spending money but also in boxing and even in living a life.)
But how can the boxer's act of spending all his money in a もったいぶらない way, or (1), relate to his boxing style? And what is a もったいぶらない boxing style? As explained above, whether something is もったいない is a subjective matter, and the boxer is a man of もったいぶらない: In many aspects, he lives his life もったいぶらないで, and he actually thinks that doing so is really nice/cool/chic! That's why he boxes もったいぶらないで, or he puts his full effort when he boxes.
Last but not least, consider the other question: Why does もったいぶらない mean unpretentious or modest, according to a dictionary? As explained above, everyone has a different sense of values: To be a もったいぶらない person, one needs to give up his/her sense of values to some extent and to be open-minded. That's why a person of もったいぶらない characteristic can be received/considered as modest, humble, generous, easy-going, and even nice/cool/chic, at least in the Japanese society.
As background information, もったいぶらず, or equivalently もったいぶらないで, is an adverb which means doing something in a not もったいぶる way. もったいぶる is a verb meaning that someone is unwilling/reluctant to do something because doing something is もったいない. We feel もったいない when we spend or miss valuable things, such as money, time, and opportunities, usually in an inappropriate way.
From the background information, we can conclude that もったいぶらず is an adverb which means doing something without hesitation/reluctance and without contemplating relevant values. In the manga, もったいぶらず is used to indicate that the boxer is very willing to spend what he has, say all his signing bonus (契約金), in order to drink in a hostess club.