The very first thing should of course be trying to tweak the instrument itself. Even if it doesn't have a dedicated “attack” control, there may be other parameters that affect the reaction time, or for orchestral sounds maybe different “playing technique” layers. Many instruments also respond faster when played in the upper dynamic range, so it could help to “play in louder” and then reduce the volume in mix.
If none of that helps, the easiest fix might be to just move the track forward a little, best done by applying “negative delay”. Most DAWs allow you to set this up, or else you can do it with various plugins. It may seem like that shouldn't be possible, but actually it is – protocols like VST have an option to signal how much delay a plugin incurs (typically because of some buffered FFT), and the DAW then automatically compensates by moving the MIDI events earlier under the hood.
Of course, when the sounds start too early, it may also be off-putting. Try first with very little anti-delay – 15 ms may be enough. If the attack is so slow that this is not enough, then the pre-fade-in would probably become untenable. You could trim this back with an additional envelope – i.e. feed the audio signal into another synth plugin that doesn't create sound of its own, only opens up a gate. This will only work if there's no reverb on the track. (But again if “chopping” is an issue you could cheat by duplicating the track and erasing every other note in each copy.)
Another approach I would consider is to leave the sound as-is, but add another faster-attacking one on top of it, feeding from the same MIDI input. There's many combinations you could try here.
Finally, you could artificially change the response of the sound you have: bounce the instrument to an audio track at half speed, then speed up that track again with a time-stretch algorithm like élastique. This will speed up that transients as well – but also any modulation in the rest of the sound; in particular sped-up vibrato can sound very silly.