This match-up I feel, as hard as it may be for some, still in Falco's favor.
Pikachu's attacks are fast, and combined with their stun, and low lag, can feel almost unpunishable for some players. Pikachu's primarily approach with full hopped jolts, or fairs. Sometimes they will QAC to reposition, but it's fairly predictable.
Because Pikachu approaches in such ways, you can shunt the jolt approach with retreating shines, and sometimes this handles the fair approach as well. Nair is a good aerial option as well if you want to react more aggressively, but if you miss your spacing, its bound to be worse for you than it will be for the rat.
The chain grabs are about even on each other in my opinion. Falco's at least leads off the edge into a spike, something which thankfully Pikachu does not have.
Killing the rat is a little more troublesome. One of Falco's most reliable kill moves is his Up smash, and Pikachu is very good at not allowing them to connect. Because of this you have to play a little more patiently and try and conserve your Bair so it's strong when you need it.
There are several strong points to this match-up though.
First off, if you are good at tap DI (what you use to get out of the down smash), then you can use it on Fair and Bair as well to improve your ability to escape Pikachu.
A lot of Pikachu's will try to combo into thunder, if you can reliably bait this, you can shine right back into them. It won't generally kill, but it will force them to be more careful in its use through out the match.
Pikachu has a lot of punishable moves, namely f-smash and d-smash when they miss. You need to react quick and punish accordingly, as many good Pikachu's won't leave themselves open many times.
Also, more so in this match-up than most, you want to avoid using your shield a lot. Pikachu is one of the best shield pressure characters, so it may be better to take the hit and tap DI away than let him eat through your shield until it stabs. Proper spacing and good mind games can circumvent the need to shield through out most of this match. Nair, Dair, Shine, Lasers, and F-tilt are all good at keeping Pikachu at the ranges you want him.