This is an interesting question.
Requirement to "Qualify"
The relevant provision seems to be subsection (b) of the Presidential Succession Act of 1947 (codified at 3 USC 19. That subsection provides that:
(b) If, at the time when under subsection (a) of this section a Speaker is to begin the discharge of the powers and duties of the office of President, there is no Speaker, or the Speaker fails to qualify as Acting President, then the President pro tempore of the Senate shall, upon his resignation as President pro tempore and as Senator, act as President.
Thus a Speaker would not become acting President unless the Speaker "qualifies". I believe this would include fulfilling all the requirements for a President, including the minimum age, and the requirement that the President be a "natural born citizen".
Subsection (d)(1) further provides that:
If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is no President pro tempore to act as President under subsection (b) of this section, then the officer of the United States who is highest on the following list, and who is not under disability to discharge the powers and duties of the office of President shall act as President: Secretary of State, Secretary of the Treasury, Secretary of Defense, Attorney General, Secretary of the Interior, Secretary of Agriculture, Secretary of Commerce, Secretary of Labor, Secretary of Health and Human Services, Secretary of Housing and Urban Development, Secretary of Transportation, Secretary of Energy, Secretary of Education, Secretary of Veterans Affairs, Secretary of Homeland Security.
The Act has never been invoked, nor have either of the two prior Presidential Succession Acts, so there is no case law or precedent available.
The Wikipedia article on the Act describes the history of all three versions, and the times when an invocation might have occurred.
Questioned Constitutionality
The Wikipedia article quotes a statement by Judge M. Miller Baker made during a September 2003 joint hearing before the U.S. Senate's Committee on Rules and Administration and Committee on the Judiciary. This statement argues that the Act as currently written is unconstitutional. The statement reads:
The 1947 Act is probably unconstitutional because it appears that the Speaker of the House and the President pro tempore of the Senate are not "Officers" eligible to act as President within the meaning of the Succession Clause. This is because in referring to an "Officer", the Succession Clause, taken in its context in Section 1 of Article II, probably refers to an "Officer of the United States", a term of art under the Constitution, rather than any officer, which would include legislative and state officers referred to in the Constitution (e.g., the reference to state militia officers found in Article I, Section 8). In the very next section of Article II, the President is empowered to "require the Opinion, in writing, of the principal Officer in each of the executive Departments" and to appoint, by and with the advice and consent of the Senate, "Officers of the United States". These are the "Officers" to whom the Succession Clause probably refers. This contextual reading is confirmed by Madison's notes from the Constitutional Convention, which reveal that the Convention's Committee of Style, which had no authority to make substantive changes, substituted "Officer" in the Succession Clause in place of "Officer of the United States", probably because the Committee considered the full phrase redundant.[1]
Since the Act has never been invoked, this contention has never been tested in a court. Given the 25th Amendment's provisions for filling a vacancy in the office of Vice-President, the likelihood of the act ever being invoked is much reduced.
Notes
[1]: "Ensuring the Continuity of the United States Government: The Presidency". Prepared Statement of M. Miller Baker, Joint Hearing Before the Committee on Rules and Administration and the Committee on the Judiciary, United States Senate. September 16, 2003. Archived from the original on January 14, 2021. Retrieved July 11, 2018 – via GlobalSecurity.org. [This citation is copied from the Wikipedia article.]