I usually follow your line (accepting the pawn), but then play 4.e3 to simply return the pawn (as the Black Knight will capture on e5). Simply returning the pawn avoids the Budapest complications while this e3 sideline actually has a decent amount of theory if you want too (but not extremely important before playing this line). Clearly learning theory is best, but this sideline of e3 avoids most "traps" of the Budapest, while simply getting into a "playable position". Keep in mind that some call this opening "Budapest Defense" not "Budapest Gambit" as it is still solid for Black - and should be taken seriously.
How to play against Budapest Gambit

For the main line, you need to play Bf4 and Nf3 (ideally in that order). Now just expect the Bishop check (Nc3 gives you doubled pawns, I recommend you get familiar with the "trappy" Nd2 line instead). From there you should be golden! Play good moves from here and you will win.
More than 200cps is certainly not solid.
There are alternatives, of course, but most of them will lead to the same result, no doubt about it.
Again, I never claim anything I have not checked thoroughly before.
And thoroughly means checking each and every line.

Certainly..... perhaps we should follow the rules of Romantic Era while playing against this gambit...... "Never decline a gambit".

Certainly..... perhaps we should follow the rules of Romantic Era while playing against this gambit...... "Never decline a gambit".
True, I always accept a gambit. If your opponent blunders a pawn on purpose you should always take it. Don't complain after the game: Boohoo I could have won a pawn in the opening!
The Alekhine variation doesn't refute the gambit, I believe it fell out of favor after Alekhine had some bad losses or draws in it (not entirely sure). Black plays the Alekhine variation by trying to undermine white's center in hypermodern fashion. I recommended it as it is a good way for an aggressive player to gain the initiative, and I enjoy playing this variation myself. Strong masters and theory says that Bf4 is the best way for white to get an advantage in the opening though. The wikipedia page for the Budapest defense also talks about some of the key ideas by each side in the opening, should be a decent supplement to the other material. https://en.wikipedia.org/wiki/Budapest_Gambit
Yeah, modern theory is mostly weak and things fall out of favour.
The very same will happen with Bf4.
e4 and f4 is simply much stronger, maybe by at least half a pawn, more or less the line between a win and a draw.
There is nothing to compensate for that powerful pawn center; activity is just a temporary phenomenon and will soon subside.
White has no weaknessess, so I guess there are 90% chances this is won for white.

Guys it's quite a strange situation to confront with...... As my topic is on playing against Budapest Gambit, from the provided resources it seems Budapest Gambit is completely refuted and it is impossible to play with black pieces.....
SF reaches 200cps also after Ng6 retreat.
I bet the Budapest is unplayable.
It is a pity so many books written on it, and no one mentions the gambit loses by force.
Top engines have changed everything, chess is not the same as it used to be 10 years ago.
One more.
With imperfect play, SF still wins.
The white advantages, the powerful pawn center, are permanent, the black activity subsides.
It takes time, but white will always be on top.

Most interesting fact about Budapest is I think the early rook manoeuvre by pushing a5 followed by Ra6 and a kingside shift..... What a brilliant and aggressive way to play. White literally can do nothing to stop that..... And one more thing is that Mamedyarov beated Kramnik using Budapest Gambit which certainly supports the fact despite achieving a good position white can lose...... Certainly it's the most fascinating way to play against d4 & c4 setup.
The Budapest isn't refuted by modern theory, it's just correct play leads to a good (not winning) position for white.
Wanna do a full analysis now?
It is your move on the diagrammed position, after f4.
I bet, in 10 moves' time, with perfect play, white will be leading +100cps.
What people don't understand is human and computer/top engine chess are COMPLETELY DIFFERENT.
That is why they rate top engines at 3300+.
Some lines, I would say close to 50%, especially involving tactics, would NEVER be seen by any normal human.
Even if he analyses a lot, but does NOT rely fully on the engine, when necessary.
Current theoreticians are STILL quoting human OTB games, in their investigations.
Look at Amazon books on openings, it is all references to human OTB games, NO top engine games at all, NO top engine analysis.
They are still old-fashioned, all of them.
In that way, QUITE OFTEN, the best line of play is missed and so the theoreticians arrive at the wrong conclusions.
As said, there are lines no human will see.
That is why I wrote 'The Secret of Chess': https://www.amazon.com/Secret-Chess-Lyudmil-Tsvetkov-ebook/dp/B074M85CVV/ref=sr_1_2?s=digital-text&ie=UTF8&qid=1521751444&sr=1-2&dpID=41Yt4-vobwL&preST=_SX342_QL70_&dpSrc=srch
to alert people of new developments, but most are just making a laughing stock of me.
You will be surprised how much theory will change in the future.
Clearly Tsvetkov has not heard of the hypermodern school of chess.
Clearly he has.
Smerdon even calls my approach 'post-modern' in his review.
Post-modernism means something new has been discovered.
I have discovered MANY new things.
BUT, on a par with that, the old principles are ALWAYS valid.
The pawn center in the middlegame can hardly be underestimated.
I have added the understanding of compactness to this pawn center, which allows me to see correctly or better a wide range of positions previously misunderstood.
So, Tarrasch and Steinitz, the pawn center, are the old school, me and the compactness of the pawn center the hypermodern.
Nimzovich was hypermodern, because he differed from the old routine.
Now I differ from Nimzovich and the rest.
Nimzovich is the classic, I am the hypermodern.

oh forgot , he refuted Modern defense too recently (and Alekhine's defense a few months before).
So you're saying he's close to solving chess?
The Alekhine Variation is best and gives white big, probably winning advantage.
Another busted opening.
Too bad that the top players in the world don't know this.