repeater or inline booster for CAT5e

A switch is a repeater - but you'd need 3 of them to cross 1000 feet and they would, presumably, be outside.  There may be PoE switches you could use... not sure.

Frankly, running copper outside is dangerous.  I've seen more than my share of destroyed equipment due to lightning strikes.  Save yourself time, money, and aggravation - put in fiber - the limit to fiber is MUCH higher (on the order of miles) and you can get 10/100 transceivers fairly cheaply.  (Gigabit would be MUCH more expensive).  

Wireless COULD be used - using directional antenna's that are aimed at each other, if you have a fairly clean line of sight - but frankly, the fiber is going to be your BEST option (but yes, also your most expensive - but I would still expect you could do this for about $1000-2000 max.).

A point to point wireless router could make the distance but you would be limited to speed.  Your best bet is fiber really.  IF you must use cat5e i would go about 250' then put in a small powered switch, and then continue off of that another 250', but that will cause a long daisy chain to the building and could cause some issues.  It would be difficult to diagnose any problems.  For a little more money fiber would be a better and faster option.  

Wireless bridge is the way to go.  This site has a complete kit for $399 and it will work up to and over 1 mile.  I'm assuming the majority of the run is outside, and if so, you won't be able to power repeaters or switches.  Also, you will need to use direct burial cable or bury conduit ($$$).  Do the wireless bridge thing.  It's easy to set up and doesn't cost too much.

Have you considered Poweline Ethernet converters? As long as you get the right Phse / Circuit it's quick & easy.

Fiber runs aren't a DIY install with typical tools.  I have everything for electrical, Cat3/5/6, coax, etc.  But, no tools or testers for fiber.  The epoxy and oven requirements were the first clue for me that it was not a one-time thing for running to the garage.

That being said, Cat5/6 is the wrong tool for this job.  1000ft would require a few switches in-line.  If buried, that would mean underground vaults that have power provided (110VAC).

Also, with 1000ft between buildings, it's safe to assume that these structures do not share the same electrical grounding (earthing) system.  That makes copper a big no-go.  In most places it is against code.  In any situation (legal or not) it's dangerous to property and person to have this conductor carrying stray current between buildings.

From a technical standpoint, I'd favor fiber.  But, I'd have to call in help, never having done it before.  If given a sufficient budget (and forgiveness for failed attempts) , I'm sure I could get the gear and a DIY project would be successful.

I'd rather bang my head against the wall a couple of times while learning how to epoxy and polish than constantly fight wireless issues over a 1-km distance.

I'm fairly certain you can get a 1000 foot Fiber cable with ends.  You have to be a little gentle with it, but otherwise, you can run it between the two buildings (buried in conduit or above ground) and connect it to two fiber transceivers, one at each end.  So splicing fiber shouldn't be necessary.

>I'm fairly certain you can get a 1000 foot Fiber cable with ends.
most definitely you can

Yes, you can get different lengths with ends.   I just don't know where....the cable supply places I go to for copper have fiber in bulk spools of 1000-, 1500-, 2000-ft or as 30-ft patch cables.

I watched my Verizon FIOS installer: he eyeballed the distance, then wrapped the extra inside the ONT box on the outside of the house.  I ended up with a few loops of extra fiber inside there, but all within the allowable bend radius.

Good solution