Chemical Reaction Alternative to O2 Respiration in H2/N2 Atmosphere
The obvious approach to is to just do the direct analog of what Earthling metabolism does: exactly reverse the photosynthetic reaction (on net, anyway; the intermediate processes may be rather different, as they are on Earth).

In this case, that means reacting hydrogen with carbohydrates to get back methane and water. Based on this paper on photosynthesis in hydrogen-dominated atmospheres, building carbohydrates from methane and water requires considerably less energy than building them from carbon dioxide and water, so we'll equally get less energy out by decomposing them again--I'd estimate somewhere around a quarter to a third as much power density as Earthling animals relying on aerobic respiration, if we assume organisms in this environment will be building biomolecules optimized for this environment, rather than just copying ours. Given how much time I spend every day not eating, and not hyperventilating, I'm pretty sure that would still be enough to power complex animals; they just might need to eat more / breathe more / be more lazy than equivalent Earth animals. Even if you only get one fifth the power density (based on the Bains paper's estimate that hydrogenic photosynthesis can proceed with 20% of the energy required by oxygenic photosynthesis), that's still 7-8 alien-equivalent-of-ATPs per glucose, as opposed to the 2 ATPs per glucose provided by anaerobic respiration (cf. 38 ATPs-per-glucose for aerobic respiration).

As Andrew noted, however, land animals will have an extra energy supply available in the form of reducing N2 into ammonia--i.e., they can supplement their energy supply simply by breathing, and only really need to eat for the materials. Incidentally, autotrophs could also make use of excess ammonia production to power chemosynthesis in the absence of light.

Despite the relatively low abundance of ammonia in the atmosphere, however, it has much higher solubility in water than hydrogen does, and thus may be more easily available to aquatic organisms than free hydrogen is. That gives you a slightly modified "respiration" reaction that takes 3 formaldehyde units and 4 ammonia molecules to produce methane, water, and free nitrogen. Decomposing ammonia into hydrogen and nitrogen is endothermic, however, so this would be less energetic than hydrogen respiration. Not sure how well that would be balanced by the reduced energy needs for pumping water over the gills compared to processing enough water to get the equivalent quantity of free hydrogen. This is a Good Thing for the long-term stability of the atmosphere, as otherwise you'd expect life to just keep producing ammonia until either nitrogen or hydrogen is exhausted.

Now, that is not the most energetic set of reactions imaginable. Obviously, if they had access to oxygen, that would be way better. But, it's the only option that is consistently renewable. Other potential oxidizers, which are more oxidized than the plant and animal biomass that serves as food in this scenario, won't be produced by plant life (by definition--anything that is produced by plantlife is biomass available as food, and thus not more oxidized than itself!) and thus can only be regenerated in small quantities by geochemical processes, and will have very short environmental half-lives. E.g., if creature could eat lithium metal, they could react that with nitrogen, ammonia, water, or complex organic biomolecules and get energy out of any of those reactions--but you just aren't going to find large quantities of free lithium to power your biosphere!

For some back-of-the-envelope estimated specifics, let's initially suppose that plants are hydrogenically producing glucose and animals are hydrogenating glucose. That leads to the following metabolic reaction:

${C_6H_{12}O_6}^{174 kJ} + 12{H_2}^{0 kJ} \rightarrow 6{CH_4}^{-74.85 kJ} + 6H_2O^{-285.83 kJ}$

Which provides approximately 2338 kJ/mol of glucose, based on the heats of formation for each reaction component as listed in the NIST Chemistry WebBook; the net energetics for aerobic respiration are found just by replacing methane with ${CO_2}^{-393.51kJ}$ in that equation (diatomic oxygen, like diatomic hydrogen, conventionally has a heat of formation of 0, so that doesn't make any difference to the calculation), which gives us a theoretical output of 4250kJ/mol. (Looking up the actual empirical energy yield of aerobic respiration of glucose, we get 2880 kJ/mol--so, assume that all other energy calculations here are only good to within about a power of two!) So, if hydrogen-breathing animals can just manage to evolve a slightly more efficient energy system than we have (which seems plausible, especially if they are living at lower temperatures which permit better thermodynamic efficiency), they may even get up to a full half the power density of oxygen-breathing Earthlings. Given the seriously speculative and highly imprecise nature of these calculations, though, I'm more comfortable sticking with my earlier 1/3 estimate as a plausible ceiling.

Plants in a hydrogen-dominated atmosphere aren't necessarily going to be producing as much glucose as our plants do in our highly oxidized environment, so let's look at some other options for biomolecules that might be available for animals to eat. Suppose, for example, that plants produce acetylene as an energy storage molecule; hydrogenating that provides

$C_2H_2^{226.73 kJ} + 3H_2^{0 kJ} \rightarrow 2CH_4^{-74.85 kJ}$

376 kJ/mol of acetylene -- or about 14.5 kJ per gram. About half as much per gram as glucose.

And if we look at the above-mentioned ammonosynthesis reaction, using hydrogen and nitrogen directly from the atmosphere, we get

$3H_2 + N_2 \rightarrow 2{NH_3}^{-45.9 kJ}$

about 91 kJ/mol of nitrogen, or 5.3 kJ/gram of inhaled hydrogen/nitrogen mix. That's not a ton compared to glucose (or acetylene), but there's just so much more of it available, for cheap! After all, it takes quite a lot energy just to consume and digest food--and practically none to just breathe it in.

Also consider that, largely because of the above reaction, nitrogen is far more easily bioavailable in this world than on Earth, so maybe they use a lot more cyanide groups or even azides. Hydrogenating hydrogen cyanide gets you

${HCN}^{135.14 kJ} + 3{H_2}^{0 kJ} \rightarrow {CH_4}^{-74.85 kJ} + {NH_3}^{-45.9 kJ}$

255.75 kJ/mol, or about 9.5 kJ / gram.

Simply decomposing hydrogen azide, even before you consider the energy gains from hydrogenation, gets you

$2{HN_3}^{264 kJ} \rightarrow {H_2}^{0 kJ} + 3{N_2}^{0 kJ}$

about 6 kJ/gram.

Those last few reactions (somewhat surprisingly, given the notorious explosivity of azides) would tend to drive the average energy density of food down a smidge from the optimistic third-to-half estimates, so once again about a quarter of the total energy density of aerobically-metabolized Earthling food seems reasonable.