Specifically, I'm looking for an adjective that describes someone who is aging and realizes there are more days behind them than in front of them. Ideally, the adjective would have a positive connotation to it rather than a negative one. Maybe I'm asking too much? lol
"a word that describes someone who is aging and realizes there are more days behind them than in front of them"
I suggest you use "over-the-hill" or "past one's prime". - These are phrasesl, though.
"no longer able to do something at an acceptable level because of age" TFD
old and no longer in the best part of your life" IM
past one's prime
Once you turn 60, you are over the hill. It's all aches and pains.
Barbra was a wonderful singer, but she's past her prime now.
- Your car is past its prime. It's hight time you bought a new one.
- She was a great actress once, but she is past her prime now.
As single-word adjectives: aged, retired, declining, etc.
The first word that comes to my mind is senescence, or, its adjective form, senescent:
the state of being old; the process of becoming old (http://www.merriam-webster.com/dictionary/senescence)
Wikipedia has a more technical definition:
Senescence (/sɪˈnɛsəns/) (from Latin: senescere, meaning "to grow old," from senex) or biological aging (also spelled biological ageing) is the gradual deterioration of function characteristic of most complex lifeforms, arguably found in all biological kingdoms, that on the level of the organism increases mortality after maturation. (https://en.m.wikipedia.org/wiki/Senescence)
You could perhaps also use the related term senile:
showing a loss of mental ability (such as memory) in old age (http://www.merriam-webster.com/dictionary/senile)
Finally, you could also consider that this person is aware of their own mortality:
the quality or state of being a person or thing that is alive and therefore certain to die; the quality or state of being mortal (http://www.merriam-webster.com/dictionary/mortality)
I hope this is helpful for you!
The Oxford dictionary of American English defines short-timer as
A person nearing the end of their period of military service.
In my experience, this is routinely used in the civilian world as well, to refer to somebody who has submitted a resignation (or announced a retirement date), but who is still at work (for a last few days).
The Cambridge English Dictionary defines lame duck as
a person who still has time to serve in an elected position despite not being elected again in a recent election, with the result that the person has no real power
Of course there are many terms for an old person (politically correct: senior citizen / slang: geezer). I can’t think of a word that encompasses both concepts. You might want to look up bucket list and last gasp and see whether they lead you anywhere.
Perhaps waning or the phrase on the wane
to draw near the end.
The year is waning, and what have we accomplished?
His childhood was waning; he would soon become an adolescent.
The OP prefers a word with a positive connotation. The positive word that is associated with aging is wiser, as in older and wiser.
Vivian Clayton, a geriatric neuropsychologist in Orinda, Calif., developed a definition of wisdom in the 1970s, when she was a graduate student, that has served as a foundation for research on the subject ever since. After scouring ancient texts for evocations of wisdom, she found that most people described as wise were decision makers. So she asked a group of law students, law professors and retired judges to name the characteristics of a wise person. Based on an analysis of their answers, she determined that wisdom consists of three key components: cognition, reflection and compassion.
The article goes on to discuss how these three components change with age. Although cognitive functioning slows with age, the quality of information in the older brain is more nuanced. This can form the basis for wise behavior.
The article concludes with a paragraph that directly addresses the OP's question:
Dr. Clayton says there’s a point in life when a fundamental shift occurs, and people start thinking about how much time they have left rather than how long they have lived. Reflecting on the meaning and structure of their lives, she said, can help people thrive after the balance shifts and there is much less time left than has gone before.
As for the aspect of running out of time to complete a task, which was in the OP's title, the older and wiser person will realize that many tasks are unimportant, and will focus on essential tasks.