Mandarin, and perhaps the entire Sino-tibetan family doesn't have adpositions. This is because they make use of 'coverbs' (NOT to be confused with CONVERBS). For instance, 'to go' doubles as 'to', 'to take' doubles as 'with(instrumental)', etc... There is the coverb for 'from' though that is rarely used as a main verb, so some people argue its the only adposition in the language. Natlangs aren't as consistent as conlangs.
Really, there's many alternate strategies to minimize adpositions. Tok Pisin for example takes this to an extreme, having only two preposotions. One means of/from, the other means everything else. The verb itself disambiguates which is meant, similar to how Spanish 'a' can be mean either 'at' or 'to' depending on the verb in question. Tok Pisin speakers also make liberal use of prepositional phrases. Instead of saying 'into the house', they say 'to the inside of the house'. Similar to how English doesn't have a dedicated preposition for 'to the right of'. A lot of languages do this when they don't have a an appropriate locational adposition. Finnish uses this when none of their cases fit, and Japanese uses this with seemingly all locationals outside of the core concepts of 'at', 'to', and 'from'.
Regarding coverbs, its not unusual for people to use verbal phrases where you may find an adposition in another. For instance, Spanish doesn't have a word for 'despite', so instead they say 'a pasar de'. In English, you can even see people say 'excluding' in place of 'without'.
As for applicatives, I believe Tagalog relies purely on applicatives. This does mean though that no verb can have more than 3 arguments, including adjuncts. Though one of its three slots is used for locationals only; all other possible arguments use either the 'direct' or 'indirect' slots.
Given this, in theory you could also just use a handful of cases, and use other constructions to convey everything else. Assuming you having cases like locatives and latives, you can even use your own form of adpositional phrase by combining it with a genitive noun (such as 'house-of inside-to'). This is why so many languages have adpositions that for some mysterious reason govern the genitive.
There's often multiple strategies to accomplish something. There aren't too many alternatives to adpositions, but there are. Of course, it seems to be rare. Even if a language lacks adpositions, alternate strategies will inevitably get whittled down to a simpler particle over time simply because of how frequently they're used. If you say things like 'house-of inside-to', expect 'inside-to' to get reduced into a singular morpheme over time.
