This depends in part on what the CLA requires, but it may be possible.
Do you have the ability to reuse the submission at all?
In general, it's understood that contributions to an open source project are offered under the same license as the project at large. In particular, the text of Apache License 2.0 makes this explicit:
- Submission of Contributions. Unless You explicitly state otherwise, any Contribution intentionally submitted for inclusion in the Work by You to the Licensor shall be under the terms and conditions of this License, without any additional terms or conditions. [...]
and
"Contribution" shall mean any work of authorship, including the original version of the Work and any modifications or additions to that Work or Derivative Works thereof, that is intentionally submitted to Licensor for inclusion in the Work by the copyright owner or by an individual or Legal Entity authorized to submit on behalf of the copyright owner [...]
Since this pull request fits the definition of a Contribution (a modification of the original work, intentionally submitted to the copyright holder for inclusion in the work), the terms of the Apache license section 5 have caused this pull request to be licensed by its author under Apache 2.0 as well. Therefore, you are entitled to use the content of the abandoned pull request in your own pull request, under the terms of the Apache 2.0 license. This is roughly the same as making a contribution that includes some borrowed Apache-licensed code from elsewhere (only in this case, the borrowed code is 100% of the submission), which is to say it's legally okay.
Can you sign your own CLA and submit someone else's code?
Most likely, the CLA is in place to deflect any legal complication away from the project back to the contributors. In particular, the CLA probably requires you to assert that your contribution does not infringe anyone's copyright, and that if it turns out that it does, that's your problem, not theirs. This could bite you if it turns out that the original pull request included infringing material from some other place. If you are comfortable assuming legal responsibility for the non-infringing status of this contribution, then you may submit it yourself under a CLA that you sign.
If the CLA requires that you assert genuine copyright ownership of your submission, because the maintainer needs you to agree to the possibility of relicensing -- something like, "I agree that my submission may be offered by the repo owner under different licensing terms in the future, like GPLv3, instead of its current Apache license" -- then you can't agree to that, because your right to reuse the pull request comes from the Apache 2.0 grant, not your original copyright ownership.
