How to bind AltGr+key to a symbol

It maps to Control_R as that's how it's configured in XKB symbols for CTRL. Changing the configuration should result in Alt_R being mapped to Control_L. Note that with this method, your custom configuration will be overwritten by any future upgrades of xkeyboard-config (at least that's the package that owns /usr/share/X11/xkb/symbols/ctrl in Archlinux).
Open /usr/share/X11/xkb/symbols/ctrl, scroll down to this section:

// right alt functions as another ctrl key
partial modifier_keys 
xkb_symbols "ralt_rctrl" {
    replace key <RALT> { type[Group1]    = "TWO_LEVEL",
                         symbols[Group1] = [ Control_R, Control_R ] };
    modifier_map Control { <RALT> }; 
};

and replace Control_R with Control_L so that it reads:

// right alt functions as another ctrl key
partial modifier_keys 
xkb_symbols "ralt_rctrl" {
    replace key <RALT> { type[Group1]    = "TWO_LEVEL",
                         symbols[Group1] = [ Control_L, Control_L ] };
    modifier_map Control { <RALT> }; 
};

save & restart X then run:

setxkbmap -option ctrl:ralt_rctrl

check with xmodmap:

xmodmap -pke | grep 108
keycode 108 = Control_L Control_L Control_L Control_L

To make it permanent add setxkbmap -option ctrl:ralt_rctrl to your session start-up.
Alternatively, add ctrl:ralt_rctrl to your xorg.conf.d config files, e.g.:

Section "InputClass"
    Identifier         "Keyboard Defaults"
    MatchIsKeyboard    "yes"
    Driver             "evdev"
    Option             "XkbLayout" "us"
    Option             "XkbRules"  "evdev"
    Option             "XkbOptions" "ctrl:ralt_rctrl"
EndSection

Note to Gnome users: Gnome overrides xorg XKB options so one has to add ctrl:ralt_rctrl via gsettings (or dconf-editor):

gsettings set org.gnome.desktop.input-sources xkb-options "['ctrl:ralt_rctrl']"

The problem is that when you use a common modifier like AltGr, programs actually see it as being pressed when you use the AltGr arrows. So Shift + AltGr + i actually appears to a program as if you are pressing Shift + Alt + Up and it's the addition of the extra Alt that confuses the program. You can avoid this by either adding more shortcuts to the program if possible, so Shift + Alt + Up does the same action as Shift + Up, or you can use an uncommon modifier that the program is not aware of.

There are other ways of doing this, but this was one of the simpler ways and works well for me. It creates a "Group2" overlay on the alphanumeric keys, such that when you switch to Group2 you will have arrows there instead of letters. You can switch to Group2 with a toggle (on/off like Caps Lock normally works) or a momentary (press to activate, like Shift normally works)

partial alphanumeric_keys
xkb_symbols "alpha_arrows" {
    key <AD07> {
        symbols[Group2]= [            Home ]  # u
    };
    key <AD08> {
        symbols[Group2]= [              Up ]  # i
    };
    key <AD09> {
        symbols[Group2]= [             End ]  # o
    };
    key <AD10> {
        symbols[Group2]= [           Prior ]  # p
    };
    key <AC07> {
        symbols[Group2]= [            Left ]  # j
    };
    key <AC08> {
        symbols[Group2]= [            Down ]  # k
    };
    key <AC09> {
        symbols[Group2]= [           Right ]  # l
    };
    key <AC10> {
        symbols[Group2]= [            Next ]  # ;
    };
    key <AB06> {
        symbols[Group2]= [          Delete ]  # n
    };
};

You can then pick a key to use as a Group2 activation key. You can see all the options (if you don't want to roll your own) with this:

$ grep grp /usr/share/X11/xkb/rules/base.lst

Then pick an option, for example this one activates the arrows only while Caps Lock is being held. It also adds a new combination Alt + Caps Lock to actually switch caps lock on and off:

$ setxkbmap -option grp:caps_switch

You can add this into your Xorg config file to make the change permanent - there are plenty of guides online about how to do this.

Note that:

• key is used, as opposed to replace key because we want to add to the existing definition, not replace it.
• If you actually have an alternate keyboard layout (possibly some international keyboards that can also type Latin/English letters), this method may not work for you. You will have to define a new type based on ALPHANUMERIC with extended modifiers (e.g. Hyper) to activate the arrow keys.