This hadith does not specifically show that Sulaiman (AS) had flying horses since the Prophet (SAW) didn't say anything to confirm it.
However, Sulaiman (AS) was given much more amazing powers than just flying horses. Allah says in the Quran:
And to Solomon [We subjected] the wind - its morning [journey was that of] a month - and its afternoon [journey was that of] a month, and We made flow for him a spring of [liquid] copper. And among the jinn were those who worked for him by the permission of his Lord. And whoever deviated among them from Our command - We will make him taste of the punishment of the Blaze. (34:12)
He had control of the wind and control of the Jinn!
Furthermore, under his command were beings who could carry a throne thousands of kilometers in the blink of an eye:
Said one who had knowledge from the Scripture, "I will bring it [the throne] to you before your glance returns to you." And when [Solomon] saw it placed before him, he said, "This is from the favor of my Lord to test me whether I will be grateful or ungrateful. And whoever is grateful - his gratitude is only for [the benefit of] himself. And whoever is ungrateful - then indeed, my Lord is Free of need and Generous." (27:40)
I would not be surprised at all even if Sulaiman (AS) was given horses that could fly because he was given things much more amazing than that. It is an extremely weak argument critics of Islam use if they use this. Allah is all-Powerful and He certainly does not lack the ability to create flying horses and much more than that.