Given a user_id, how do I find the username?

You can access it without needing API via internal /user/ endpoint instagram uses for AJAX requests:

https://i.instagram.com/api/v1/users/{user_id}/info/

where {user_id} is numeric user id like 6817966272.

Returned response example (see user['username'] key for username):

GET https://i.instagram.com/api/v1/users/6817966272/info/

{
  "user": {
    "pk": 6817966272,
    "username": "myriaamaa",
    "full_name": "\u2661",
    "is_private": false,
    "profile_pic_url": "https://instagram.fcnx2-1.fna.fbcdn.net/vp/66486d198fc02046d04d7bc11e51e54a/5D913015/t51.2885-19/s150x150/61226760_298998544311382_2345929352031502336_n.jpg?_nc_ht=instagram.fcnx2-1.fna.fbcdn.net",
    "profile_pic_id": "2056076981860037983_6817966272",
    "is_verified": false,
    "has_anonymous_profile_picture": false,
    "media_count": 216,
    "follower_count": 4926,
    "following_count": 83,
    "following_tag_count": 0,
    "biography": "YOU. ARE. HOLY \ud83c\udf19",
    "external_url": "",
    "total_igtv_videos": 0,
    "total_ar_effects": 0,
    "usertags_count": 6,
    "is_favorite": false,
    "is_interest_account": true,
    "hd_profile_pic_versions": [
      {
        "width": 320,
        "height": 320,
        "url": "https://instagram.fcnx2-1.fna.fbcdn.net/vp/fafecdc76c82de85580c9c03d14b1aaa/5D9BD2E5/t51.2885-19/s320x320/61226760_298998544311382_2345929352031502336_n.jpg?_nc_ht=instagram.fcnx2-1.fna.fbcdn.net"
      },
      {
        "width": 640,
        "height": 640,
        "url": "https://instagram.fcnx2-1.fna.fbcdn.net/vp/0ec5339e3958c9c41414e5378fa2443c/5D7DD28A/t51.2885-19/s640x640/61226760_298998544311382_2345929352031502336_n.jpg?_nc_ht=instagram.fcnx2-1.fna.fbcdn.net"
      }
    ],
    "hd_profile_pic_url_info": {
      "url": "https://instagram.fcnx2-1.fna.fbcdn.net/vp/8b3859950f0bb8e1a4a8f65566992b78/5D9132EF/t51.2885-19/61226760_298998544311382_2345929352031502336_n.jpg?_nc_ht=instagram.fcnx2-1.fna.fbcdn.net",
      "width": 774,
      "height": 774
    },
    "mutual_followers_count": 0,
    "has_highlight_reels": true,
    "can_be_reported_as_fraud": false,
    "is_business": false,
    "account_type": 1,
    "is_call_to_action_enabled": null,
    "include_direct_blacklist_status": true,
    "is_potential_business": true,
    "is_bestie": false,
    "has_unseen_besties_media": false,
    "show_account_transparency_details": false,
    "auto_expand_chaining": false,
    "highlight_reshare_disabled": false
  },
  "status": "ok"
}

EDIT 2019-10:

This endpoint only serves user-agents that identify themselves as instagram app.

I got it working with iphone's instagram user agent:

Mozilla/5.0 (iPhone; CPU iPhone OS 10_3_3 like Mac OS X) AppleWebKit/603.3.8 (KHTML, like Gecko) Mobile/14G60 Instagram 12.0.0.16.90 (iPhone9,4; iOS 10_3_3; en_US; en-US; scale=2.61; gamut=wide; 1080x1920)

For more check out this online user agent database: https://developers.whatismybrowser.com/useragents/explore/software_name/instagram/1

# top 5:
Mozilla/5.0 (Linux; Android 8.1.0; motorola one Build/OPKS28.63-18-3; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/70.0.3538.80 Mobile Safari/537.36 Instagram 72.0.0.21.98 Android (27/8.1.0; 320dpi; 720x1362; motorola; motorola one; deen_sprout; qcom; pt_BR; 132081645)  
Mozilla/5.0 (Linux; Android 7.0; Lenovo K33b36 Build/NRD90N; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/65.0.3325.109 Mobile Safari/537.36 Instagram 41.0.0.13.92 Android (24/7.0; 480dpi; 1080x1920; LENOVO/Lenovo; Lenovo K33b36; K33b36; qcom; pt_BR; 103516666)  
Mozilla/5.0 (Linux; Android 5.0.1; LG-H342 Build/LRX21Y; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/65.0.3325.109 Mobile Safari/537.36 Instagram 40.0.0.14.95 Android (21/5.0.1; 240dpi; 480x786; LGE/lge; LG-H342; c50ds; c50ds; pt_BR; 102221277)  
Mozilla/5.0 (Linux; Android 6.0.1; SM-G610M Build/MMB29K; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/65.0.3325.109 Mobile Safari/537.36 Instagram 41.0.0.13.92 Android (23/6.0.1; 480dpi; 1080x1920; samsung; SM-G610M; on7xelte; samsungexynos7870; pt_BR; 103516666)  
Mozilla/5.0 (Linux; Android 8.0.0; SM-A520F Build/R16NW; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/66.0.3359.158 Mobile Safari/537.36 Instagram 46.0.0.15.96 Android (26/8.0.0; 480dpi; 1080x1920; samsung; SM-A520F; a5y17lte; samsungexynos7880; pt_BR; 109556226)

Edit 2020-05/2022-05: Currently this endpoint returns less information but username is still present:

{
    "user": {
        "username": "meram.ba",
        "pk": 6817966272,
        "profile_pic_url": "https://instagram.fbkk5-5.fna.fbcdn.net/v/t51.2885-19/s150x150/93795245_228684278392964_3931284463400517632_n.jpg?_nc_ht=instagram.fbkk5-5.fna.fbcdn.net&_nc_ohc=O6-_BTmMHUIAX9Qy8up&oh=8303e6fc8444b920c0c0edc603c917b5&oe=5EE91CF3"
    },
    "status": "ok"
}

If you don't have approved instagram application I suggest to use this php library: https://github.com/postaddictme/instagram-php-scraper

$instagram = Instagram::withCredentials('username', 'password', 'path/to/cache/');
$account = $instagram->getAccountById('193886659');
echo $account->getUsername();

Or direct access:

https://www.instagram.com/query/?q=ig_user(3){id,username,external_url,full_name,profile_pic_url,biography,followed_by{count},follows{count},media{count},is_private,is_verified}

Update: This url doesn't work anymore. You need to use POST. See repo to know how