So did Alphys inject Undyne with Determination in the neutral roue fight?
No. She just watched enough anime to know what DETERMINATION looks like when humans use it, then used the DETERMINATION that she generated to hold herself together for a bit longer.
She can probably just generate on her own. Boss Monsters souls can maintain themselves for a short period beyond death, so they must have some amount of determination in them. Not a stretch to go from that to thinking Undyne just creates determination.