There are several possible ways to get there, but the answer is "there is no such position."
Acting President vs. President
Under the Constitution:
In case of the removal of the President from office, or of his death, resignation, or inability to discharge the powers and duties of the said office, the same shall devolve on the Vice President, and the Congress may by law provide for the case of removal, death, resignation or inability, both of the President and Vice President, declaring what officer shall then act as President, and such officer shall act accordingly, until the disability be removed, or a President shall be elected.
When William Henry Harrison died, there was some question as to what "the same" that devolved on Tyler (his VP) meant. Some thought it meant that Tyler was Acting President; others thought the office itself went to Tyler. Tyler was in the latter group, and set a precedent that the VP became the President (finishing out the President's term) if the President died; the VP wasn't just acting. The 25th Amendment later clarified that presidential death, removal, or resignation made the VP the President; if the President was unable to exercise the powers and duties of the office, the VP was Acting President.
Further down the line, the Constitution explicitly says Congress decides "who shall act as President." Congress only talks about succession further down the line in terms of the Acting President and the officer acting as President. So, if that interpretation is correct, the answer is "the VP is the only officer who can assume the office of President in the event of the President's death." Of course, a new precedent could potentially be set if this situation ever arises; if someone will definitely act as President until the end of the term, then it makes a certain amount of sense to say they just assume the office itself.
Eligibility for office
The Constitution doesn't say "you can't be elected President unless over 35, natural-born citizen, and 14 years a resident." It says you are not eligible for the office unless you meet those requirements. That would include any way of assuming the office, including succession. You can't become President if you are not eligible to hold the office, period (that's what eligibility for an office means); the only requirements you'd get around are those covering eligibility to be elected.
It's unclear whether the Constitution's eligibility requirements apply to a person acting as President who does not assume the office. However, Congress has decided that they should. 3 USC §19, which sets out who acts as President if both President and VP are unable to, says:
(e) Subsections (a), (b), and (d) of this section [i.e. the ones listing people who can act as President] shall apply only to such officers as are eligible to the office of President under the Constitution.