All of the advice above is good advice.
Hard shifting could be caused by a bad shift box, a bad or kinked shift or throttle cable (yes a bad throttle cable can cause hard shifting), the shift shafts (as your dealer suggests), or the lower unit.
The shift cable gets kinked easily if you raise the engines all the way up and the cable does not have ample clearance in front of the engine. The kink is usually a couple of inches or so from the engine.
Hard shifting, in the 130, can be also be caused by the idle being too high. It will be especially hard to get out of gear. One way to check this is to try shifting with the engine off. You have to have someone rotate the prop until it locks into gear. If there is no change is how hard it is, it is not the idle.
When you have them show you the problem, both of the throttle and shift cables should be disconnected at the engine and the lower unit should be removed. If the lower unit is not removed, you can not really isolate the problem for sure.
If both the throttle and shift cables are disconnected, the shifter should move really easily.
If that is the case, then you should try to move the shift lever on the engine and once you move it out of the detent, it should be real easy to move. If it is really stiff the whole way, then the dealer is probably correct in their diagnosis.
If all that is easy, then it is in the lower units. Most unlikely that both lower units got hard to shift at the same time...but possible.
If they are right, here is what is going on....part #9 is what comes out of the lower case of the engine and the shift lever #13 is connected to. There could be some problems in part #13.
http://www.boats.net/parts/search/Honda/Outboard Engine/2001/BF90A1 LRTA /SHIFT SHAFT/parts.html but they are thinking it is in parts #s 9, 24, 6, and 29 and the opening that 6 goes through. Make sure they can make sure that #13 is not the problem. You do not have to remove the engine to replace that.
If this is in salt water and seals #29 fail and allow water into the flywheel housing, salt can accumulate and bind things up. Even if it is fresh water, it can happen.
As for what it takes to fix if they are right....
You have to remove the lower unit, the midsection, oil pan, watertube, lower motor mounts, etc...then you have access to the bolts that allow you to remove the powerhead. There are still hoses and wiring that have to be disconnected around the powerhead, then it will come off.
There are probably about $200 - $300 worth of parts and seals to put it back together...maybe a little more if they include changing the water pumps, spark plugs, filters, oil, etc for winterization,
Hopefully, they are planning on changing the vertical shaft seals in the lower case and will inspect the bottom of the powerhead for rust and leakage from the steel plugs in the bottom of the powerhead.
All of this takes time to do and to make sure all is checked while it is apart...now is the time to replace whatever seals they can.
If all goes well it is a full day's job to disassemble and reassemble..that does not include fixing things that they find. Not hard, but there are a lot of steps and you do not want them to rush...so nothing is forgotten.
It is hard to second guess someone else's estimates...especially on a saltwater motor. Will all the bolts come out as they should, what will break, etc?
If it takes two 8 hours days (16 hours) each at $100/hr that is $1600 plus parts. So that could put you over $2000. If their hourly rate is $120/hr, then it is $1920 plus parts...getting closer to $3000.
Hope this helps.
As Jimmy says, since they have the boat in storage, you have all winter to decide....as long as they do not raise their labor rate or are giving you the "discounted" winter rate...which goes up in the Spring. It is a lot of money to spend to improve the shifting. However, if they are right, the shifting is not going to get any better on its own.
Mike